Zero over Zero, Infinity over Infinity
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Zero Times Infinity
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Exponents Infinity over Zero, Zero over Zero, One over Infinity
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Infinity Minus Infinity
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[{"Name":"Zero over Zero, Infinity over Infinity","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"0\\0, ∞\\∞","Duration":"17m 1s","ChapterTopicVideoID":1333,"CourseChapterTopicPlaylistID":52079,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1333.jpeg","UploadDate":"2019-10-29T12:43:28.0400000","DurationForVideoObject":"PT17M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.330","Text":"This is the first of 4 parts of some theory explaining all about L\u0027Hopital\u0027s rule,"},{"Start":"00:06.330 ","End":"00:08.640","Text":"what it is, and how to use it, and so on."},{"Start":"00:08.640 ","End":"00:11.940","Text":"Question is, what is L\u0027Hopital\u0027s rule,"},{"Start":"00:11.940 ","End":"00:15.225","Text":"and how can it help us to compute the limit of a function?"},{"Start":"00:15.225 ","End":"00:21.090","Text":"Suppose I have a limit as x goes to something, let\u0027s just say,"},{"Start":"00:21.090 ","End":"00:27.525","Text":"a of the quotient of 2 functions, f^x over g^x."},{"Start":"00:27.525 ","End":"00:31.875","Text":"Let\u0027s say that after we\u0027ve tried substituting a in each of them,"},{"Start":"00:31.875 ","End":"00:35.235","Text":"we get something of the form 0 over 0."},{"Start":"00:35.235 ","End":"00:39.610","Text":"L\u0027Hopital found an extra clever and useful tool."},{"Start":"00:39.610 ","End":"00:43.670","Text":"What he said was if you have a 0 over 0,"},{"Start":"00:43.670 ","End":"00:50.419","Text":"and it turns out that what I\u0027m about to say will also work for infinity over infinity."},{"Start":"00:50.419 ","End":"00:52.375","Text":"Both of them are problematic."},{"Start":"00:52.375 ","End":"00:54.965","Text":"Then there\u0027s a shortcut trick that we can use."},{"Start":"00:54.965 ","End":"01:00.530","Text":"In this case, what he said was that if instead of taking the original limit,"},{"Start":"01:00.530 ","End":"01:03.650","Text":"we take the same x goes to a,"},{"Start":"01:03.650 ","End":"01:06.245","Text":"but instead of f^x over g^x,"},{"Start":"01:06.245 ","End":"01:08.960","Text":"we write f prime of x,"},{"Start":"01:08.960 ","End":"01:11.690","Text":"over g prime of x,"},{"Start":"01:11.690 ","End":"01:14.420","Text":"by which I mean we differentiate the top,"},{"Start":"01:14.420 ","End":"01:17.675","Text":"and separately we differentiate the bottom."},{"Start":"01:17.675 ","End":"01:20.030","Text":"This has the same answer as this."},{"Start":"01:20.030 ","End":"01:24.450","Text":"Supposing I did this and it came out to be 5,"},{"Start":"01:24.450 ","End":"01:28.105","Text":"then the original limit is also 5."},{"Start":"01:28.105 ","End":"01:31.220","Text":"If I summarize briefly what I said."},{"Start":"01:31.220 ","End":"01:34.160","Text":"If we have a limit of a quotient,"},{"Start":"01:34.160 ","End":"01:40.235","Text":"which turns out to be 1 of those 0 over 0 cases or infinity over infinity cases, then,"},{"Start":"01:40.235 ","End":"01:42.200","Text":"instead of computing this limit,"},{"Start":"01:42.200 ","End":"01:44.750","Text":"we compute a fresh limit,"},{"Start":"01:44.750 ","End":"01:50.570","Text":"which is obtained by differentiating the numerator and denominator separately."},{"Start":"01:50.570 ","End":"01:53.195","Text":"Then the answer to this question,"},{"Start":"01:53.195 ","End":"01:56.480","Text":"this limit, will be the same as the answer to the original."},{"Start":"01:56.480 ","End":"02:00.680","Text":"The best thing to do now would be to give some examples."},{"Start":"02:00.680 ","End":"02:03.230","Text":"Now, if we were to start doing this,"},{"Start":"02:03.230 ","End":"02:07.475","Text":"and we would first of all try and see if substitution works."},{"Start":"02:07.475 ","End":"02:09.965","Text":"We try putting x equals 1."},{"Start":"02:09.965 ","End":"02:12.724","Text":"Now in our heads, we could say that the denominator,"},{"Start":"02:12.724 ","End":"02:14.270","Text":"1 squared plus 1,"},{"Start":"02:14.270 ","End":"02:17.620","Text":"minus 2, which is 1 plus 1 minus 2,"},{"Start":"02:17.620 ","End":"02:19.325","Text":"2 minus 2 is 0,"},{"Start":"02:19.325 ","End":"02:21.185","Text":"and a 0 in the bottom."},{"Start":"02:21.185 ","End":"02:24.220","Text":"If we tried putting x equals 1 here,"},{"Start":"02:24.220 ","End":"02:27.695","Text":"we\u0027d get 4 plus 10 minus 14,"},{"Start":"02:27.695 ","End":"02:30.300","Text":"and also, it\u0027s a 0."},{"Start":"02:30.300 ","End":"02:35.355","Text":"This is 1 of those cases where it is a 0 over 0."},{"Start":"02:35.355 ","End":"02:40.970","Text":"This is our f, and this is our g. What we do"},{"Start":"02:40.970 ","End":"02:46.240","Text":"is we write that we\u0027re going to use L\u0027Hopital in the 0 over 0 case."},{"Start":"02:46.240 ","End":"02:50.000","Text":"You\u0027ll write equals at the bottom,"},{"Start":"02:50.000 ","End":"02:52.715","Text":"the L, not the top."},{"Start":"02:52.715 ","End":"02:56.000","Text":"The 0 over 0 case,"},{"Start":"02:56.000 ","End":"02:59.720","Text":"as opposed to the infinity over infinity case."},{"Start":"02:59.720 ","End":"03:08.250","Text":"Now, we copied the same limit point"},{"Start":"03:08.250 ","End":"03:11.310","Text":"for x. X goes to 1,"},{"Start":"03:11.310 ","End":"03:14.970","Text":"but we replace this whole thing."},{"Start":"03:14.970 ","End":"03:16.200","Text":"This is too complicated."},{"Start":"03:16.200 ","End":"03:18.305","Text":"Let\u0027s simplify this with L\u0027Hopital."},{"Start":"03:18.305 ","End":"03:21.425","Text":"Let\u0027s just differentiate top and bottom."},{"Start":"03:21.425 ","End":"03:26.730","Text":"We see that we get 8_x plus 10."},{"Start":"03:27.220 ","End":"03:35.180","Text":"This is 8_x plus 10."},{"Start":"03:35.180 ","End":"03:38.880","Text":"On the bottom, 2_x plus 1."},{"Start":"03:41.720 ","End":"03:44.570","Text":"Then, at this point,"},{"Start":"03:44.570 ","End":"03:47.645","Text":"we see that if we put x equals 1,"},{"Start":"03:47.645 ","End":"03:50.845","Text":"that we no longer have 0s."},{"Start":"03:50.845 ","End":"03:57.390","Text":"We get twice 1 plus 1 is 3 and 8 times 1,"},{"Start":"03:57.390 ","End":"03:59.730","Text":"plus 10 is 18."},{"Start":"03:59.730 ","End":"04:08.820","Text":"This is equal to 18 over 3,"},{"Start":"04:08.820 ","End":"04:11.685","Text":"and the answer is 6."},{"Start":"04:11.685 ","End":"04:17.810","Text":"I see how much shorter this is than any other technique we might use."},{"Start":"04:17.810 ","End":"04:22.685","Text":"What we would usually do would probably be to try and factorize this,"},{"Start":"04:22.685 ","End":"04:27.829","Text":"and that\u0027s a whole lot of work with solving a quadratic equation twice,"},{"Start":"04:27.829 ","End":"04:31.010","Text":"and then finding that 1 of those factors maybe is the same,"},{"Start":"04:31.010 ","End":"04:32.225","Text":"and so we\u0027d cancel,"},{"Start":"04:32.225 ","End":"04:33.815","Text":"and then we\u0027d substitute."},{"Start":"04:33.815 ","End":"04:35.630","Text":"This is much quicker."},{"Start":"04:35.630 ","End":"04:41.355","Text":"Let\u0027s go for another example again of the 0 over 0 variety."},{"Start":"04:41.355 ","End":"04:52.410","Text":"The example will be the following limit as x goes to 4 of square root of 2x plus 1,"},{"Start":"04:52.410 ","End":"04:58.965","Text":"minus 3 over x squared plus x minus 20."},{"Start":"04:58.965 ","End":"05:04.170","Text":"Now, once again, if we tried substituting x equals 4,"},{"Start":"05:04.170 ","End":"05:08.220","Text":"we see twice 4 plus 1 is 9."},{"Start":"05:08.220 ","End":"05:09.440","Text":"The square root of that is 3,"},{"Start":"05:09.440 ","End":"05:12.245","Text":"3 minus 3 is 0."},{"Start":"05:12.245 ","End":"05:14.385","Text":"That\u0027s 0 on the top."},{"Start":"05:14.385 ","End":"05:19.260","Text":"As for the bottom, 4 squared is 16 plus 4 is 20,"},{"Start":"05:19.260 ","End":"05:21.285","Text":"minus 20 is 0."},{"Start":"05:21.285 ","End":"05:24.835","Text":"This is another example of 0 over 0."},{"Start":"05:24.835 ","End":"05:26.985","Text":"Because that is so,"},{"Start":"05:26.985 ","End":"05:28.815","Text":"we can now write equals."},{"Start":"05:28.815 ","End":"05:32.625","Text":"Here I\u0027m going to write the 0 over 0 part."},{"Start":"05:32.625 ","End":"05:38.235","Text":"Then the initial of the fellow who invented this rule, L,"},{"Start":"05:38.235 ","End":"05:41.435","Text":"and then we write down a new limit,"},{"Start":"05:41.435 ","End":"05:45.040","Text":"which will be easier for us to solve than the old limit."},{"Start":"05:45.040 ","End":"05:46.610","Text":"It\u0027s not always the case."},{"Start":"05:46.610 ","End":"05:48.020","Text":"Sometimes we try L\u0027Hopital,"},{"Start":"05:48.020 ","End":"05:49.595","Text":"and it actually makes it worse,"},{"Start":"05:49.595 ","End":"05:51.920","Text":"but usually, it makes it better,"},{"Start":"05:51.920 ","End":"05:54.515","Text":"and, at least, it\u0027s an option that you have to try."},{"Start":"05:54.515 ","End":"05:58.170","Text":"The same x goes to 4,"},{"Start":"06:00.590 ","End":"06:06.220","Text":"only this time we differentiate top and bottom."},{"Start":"06:06.220 ","End":"06:08.120","Text":"Let\u0027s do the bottom first. It\u0027s easier."},{"Start":"06:08.120 ","End":"06:12.560","Text":"X squared plus x minus 20 gives us 2x plus 1, and that\u0027s it."},{"Start":"06:12.560 ","End":"06:15.930","Text":"Here\u0027s 2x plus 1."},{"Start":"06:15.930 ","End":"06:18.100","Text":"In the numerator, well,"},{"Start":"06:18.100 ","End":"06:20.840","Text":"the minus 3 doesn\u0027t account for anything that goes,"},{"Start":"06:20.840 ","End":"06:23.340","Text":"but what about the square root?"},{"Start":"06:23.410 ","End":"06:29.270","Text":"You may have forgotten that there is such a thing called a chain rule."},{"Start":"06:29.270 ","End":"06:30.680","Text":"This is a function of a function."},{"Start":"06:30.680 ","End":"06:32.690","Text":"It\u0027s not the square root of x,"},{"Start":"06:32.690 ","End":"06:36.930","Text":"it\u0027s the square root of something else, 2x plus 1."},{"Start":"06:36.930 ","End":"06:44.820","Text":"I\u0027ll just write at the side that if we have the square root of some expressions,"},{"Start":"06:44.820 ","End":"06:46.725","Text":"call it box, I don\u0027t know."},{"Start":"06:46.725 ","End":"06:50.735","Text":"Some expression with x and we differentiate this."},{"Start":"06:50.735 ","End":"06:55.440","Text":"What we get is,"},{"Start":"06:56.600 ","End":"06:59.970","Text":"and I haven\u0027t put the 1 there yet deliberately,"},{"Start":"06:59.970 ","End":"07:07.475","Text":"1 over twice the square root of whatever that was."},{"Start":"07:07.475 ","End":"07:13.115","Text":"But then something called the internal derivative as part of the chain rule,"},{"Start":"07:13.115 ","End":"07:15.650","Text":"we also have to write the derivative,"},{"Start":"07:15.650 ","End":"07:19.490","Text":"multiply by the derivative of what the box was."},{"Start":"07:19.490 ","End":"07:21.720","Text":"I wrote that on the top."},{"Start":"07:23.630 ","End":"07:27.040","Text":"Here\u0027s what we get."},{"Start":"07:27.770 ","End":"07:34.510","Text":"Excuse me, I just want to go back to my other color."},{"Start":"07:38.070 ","End":"07:40.450","Text":"I\u0027m putting this part in over"},{"Start":"07:40.450 ","End":"07:45.910","Text":"twice the square root"},{"Start":"07:45.910 ","End":"07:53.170","Text":"of 2x plus 1, that\u0027s this part."},{"Start":"07:53.170 ","End":"07:56.410","Text":"Then the derivative of that box,"},{"Start":"07:56.410 ","End":"08:00.820","Text":"so the derivative of 2x plus 1 is just 2."},{"Start":"08:00.820 ","End":"08:04.869","Text":"Now, notice that this case we\u0027re lucky,"},{"Start":"08:04.869 ","End":"08:10.315","Text":"the 2\u0027s cancel and after we cancel the 2\u0027s,"},{"Start":"08:10.315 ","End":"08:16.660","Text":"and canceling they\u0027re just like this f here, we get this"},{"Start":"08:16.660 ","End":"08:22.810","Text":"but a denominator in the numerator just goes to the denominator,"},{"Start":"08:22.810 ","End":"08:25.225","Text":"so I\u0027m leaving 1 here."},{"Start":"08:25.225 ","End":"08:28.165","Text":"Basically what we are left with,"},{"Start":"08:28.165 ","End":"08:32.215","Text":"if I leave as 1 on the top and this goes into the bottom."},{"Start":"08:32.215 ","End":"08:36.700","Text":"This is 1 over 2x plus"},{"Start":"08:36.700 ","End":"08:46.885","Text":"1 and I put it in brackets times square root of 2x plus 1."},{"Start":"08:46.885 ","End":"08:51.130","Text":"Now have a very simple limit because we can just substitute,"},{"Start":"08:51.130 ","End":"08:55.675","Text":"just putting x equals 4 twice 4 plus 1 is 9,"},{"Start":"08:55.675 ","End":"09:00.370","Text":"so what we get is 1 over,"},{"Start":"09:00.370 ","End":"09:04.735","Text":"this is equal to 1 over."},{"Start":"09:04.735 ","End":"09:09.385","Text":"Now this is 9, and this is square root of 9,"},{"Start":"09:09.385 ","End":"09:15.235","Text":"which is 3, so altogether the answer is 1 over 27."},{"Start":"09:15.235 ","End":"09:18.550","Text":"Sometimes we have to use L\u0027Hopital more than"},{"Start":"09:18.550 ","End":"09:21.820","Text":"once and I\u0027d like give you an example of that."},{"Start":"09:21.820 ","End":"09:24.460","Text":"If we put x equals 1 into here,"},{"Start":"09:24.460 ","End":"09:28.810","Text":"we get 1 minus 4 plus 5 minus 2 it\u0027s 0 and likewise,"},{"Start":"09:28.810 ","End":"09:31.315","Text":"if you substitute here, you\u0027ll get 0."},{"Start":"09:31.315 ","End":"09:37.090","Text":"We do have indeed 0 over 0 situation."},{"Start":"09:37.090 ","End":"09:39.625","Text":"According to L\u0027Hopital,"},{"Start":"09:39.625 ","End":"09:42.535","Text":"in this 0 over 0 case,"},{"Start":"09:42.535 ","End":"09:46.630","Text":"we can compute a different limit instead and"},{"Start":"09:46.630 ","End":"09:51.400","Text":"that\u0027s the 1 which we get when we differentiate top and bottom."},{"Start":"09:51.400 ","End":"10:02.560","Text":"At the top, we get 3x squared from here minus 8x plus 5,"},{"Start":"10:02.560 ","End":"10:06.430","Text":"and on the bottom we will get 3x squared"},{"Start":"10:06.430 ","End":"10:16.205","Text":"also minus 12x plus 9."},{"Start":"10:16.205 ","End":"10:22.060","Text":"Now again, if you substitute x equals 1 on the bottom,"},{"Start":"10:22.060 ","End":"10:27.445","Text":"we\u0027ll get 3 plus 9 minus 12 is 0 and here 3 plus 5 minus 8 is 0,"},{"Start":"10:27.445 ","End":"10:30.085","Text":"again is 0 over 0."},{"Start":"10:30.085 ","End":"10:36.040","Text":"Once again, I can say that this equals,"},{"Start":"10:36.040 ","End":"10:42.670","Text":"according to L\u0027Hopital in the 0 over 0 scenario this will"},{"Start":"10:42.670 ","End":"10:50.440","Text":"equal the limit as x goes to 1."},{"Start":"10:50.440 ","End":"10:59.140","Text":"Again, so 6x minus"},{"Start":"10:59.140 ","End":"11:00.740","Text":"8"},{"Start":"11:02.250 ","End":"11:10.855","Text":"over, see again"},{"Start":"11:10.855 ","End":"11:15.535","Text":"6x and minus 12."},{"Start":"11:15.535 ","End":"11:19.780","Text":"This time if we substitute x equals 1,"},{"Start":"11:19.780 ","End":"11:23.080","Text":"we\u0027re now finally unstuck we\u0027re added the whole 0 over"},{"Start":"11:23.080 ","End":"11:26.830","Text":"0 thing because this is 6 minus 12,"},{"Start":"11:26.830 ","End":"11:29.425","Text":"which is minus 6."},{"Start":"11:29.425 ","End":"11:32.155","Text":"This is just plain,"},{"Start":"11:32.155 ","End":"11:37.240","Text":"over here minus 6 and here 6 minus 8,"},{"Start":"11:37.240 ","End":"11:42.710","Text":"which is minus 2, which equals 1/3."},{"Start":"11:43.230 ","End":"11:46.090","Text":"That\u0027s our answer for this 1."},{"Start":"11:46.090 ","End":"11:49.810","Text":"This was an example that we had to use L\u0027Hopital twice in succession,"},{"Start":"11:49.810 ","End":"11:54.559","Text":"although it often happens that you use it 4 or 5 or even more times."},{"Start":"11:59.040 ","End":"12:06.490","Text":"We\u0027ve seen quite a few examples of the use of L\u0027Hopital\u0027s Rule in the case of 0 over 0,"},{"Start":"12:06.490 ","End":"12:13.730","Text":"so how about doing some infinity over infinity?"},{"Start":"12:14.220 ","End":"12:21.925","Text":"Before I give the example of the infinity over infinity case and use L\u0027Hopital,"},{"Start":"12:21.925 ","End":"12:24.685","Text":"we\u0027ll need some formulae,"},{"Start":"12:24.685 ","End":"12:26.515","Text":"we need to be reminded"},{"Start":"12:26.515 ","End":"12:36.535","Text":"in very common to see the exponential function and logarithmic function."},{"Start":"12:36.535 ","End":"12:38.650","Text":"You can\u0027t remember everything,"},{"Start":"12:38.650 ","End":"12:42.650","Text":"so I\u0027ll just write some formulae down here,"},{"Start":"12:43.020 ","End":"12:46.015","Text":"so here they are."},{"Start":"12:46.015 ","End":"12:48.439","Text":"Little formulae."},{"Start":"12:48.510 ","End":"12:54.040","Text":"Equalities. E to the power of infinity is"},{"Start":"12:54.040 ","End":"12:58.500","Text":"infinity and this really means that e to"},{"Start":"12:58.500 ","End":"13:03.395","Text":"the power of x when x goes to infinity is infinity."},{"Start":"13:03.395 ","End":"13:04.990","Text":"Infinity is not really a number,"},{"Start":"13:04.990 ","End":"13:06.490","Text":"but we treat it as if it was."},{"Start":"13:06.490 ","End":"13:10.610","Text":"It\u0027s actually just shorthand for limit."},{"Start":"13:10.740 ","End":"13:14.394","Text":"E to the minus infinity is 0."},{"Start":"13:14.394 ","End":"13:17.995","Text":"The natural log of infinity is infinity."},{"Start":"13:17.995 ","End":"13:24.775","Text":"The natural log of positive 0 is minus infinity."},{"Start":"13:24.775 ","End":"13:28.510","Text":"We\u0027ll be needing these as we go along and in general,"},{"Start":"13:28.510 ","End":"13:31.090","Text":"it\u0027s good to have these memorized."},{"Start":"13:31.090 ","End":"13:33.490","Text":"For the particular exercise,"},{"Start":"13:33.490 ","End":"13:39.595","Text":"I have in mind for the use of infinity over infinity in L\u0027Hopital is"},{"Start":"13:39.595 ","End":"13:47.500","Text":"the limit as x goes to infinity of e^x over x squared."},{"Start":"13:47.500 ","End":"13:52.660","Text":"Let\u0027s just substitute x equals infinity,"},{"Start":"13:52.660 ","End":"13:54.520","Text":"so e to the infinity,"},{"Start":"13:54.520 ","End":"13:56.365","Text":"I\u0027ve already written the formula here,"},{"Start":"13:56.365 ","End":"14:00.820","Text":"is infinity and infinity squared is obviously infinity."},{"Start":"14:00.820 ","End":"14:04.150","Text":"We have an infinity over infinity case here,"},{"Start":"14:04.150 ","End":"14:07.390","Text":"so what we\u0027ll do is we\u0027ll write that this equals,"},{"Start":"14:07.390 ","End":"14:13.765","Text":"according to L\u0027Hopital for the case of infinity over infinity,"},{"Start":"14:13.765 ","End":"14:21.610","Text":"the limit also x goes to infinity."},{"Start":"14:21.610 ","End":"14:26.500","Text":"But this time we derived top and bottom differentiate."},{"Start":"14:26.500 ","End":"14:29.725","Text":"For e to the x, we get e to the x."},{"Start":"14:29.725 ","End":"14:33.235","Text":"For x squared, we get 2x."},{"Start":"14:33.235 ","End":"14:36.310","Text":"Again, if we put x equals infinity,"},{"Start":"14:36.310 ","End":"14:40.090","Text":"e to the infinity is infinity twice infinity is infinity."},{"Start":"14:40.090 ","End":"14:45.250","Text":"We do another application of L\u0027Hopital\u0027s rule"},{"Start":"14:45.250 ","End":"14:49.675","Text":"and this time we get the same L for L\u0027Hopital,"},{"Start":"14:49.675 ","End":"14:53.180","Text":"the infinity over infinity case."},{"Start":"14:53.520 ","End":"14:56.605","Text":"We can replace this limit with a different limit"},{"Start":"14:56.605 ","End":"14:59.155","Text":"where we differentiate the top and bottom."},{"Start":"14:59.155 ","End":"15:07.720","Text":"We get the limit as x goes to infinity of e^x over 2 and this time,"},{"Start":"15:07.720 ","End":"15:15.805","Text":"we can actually substitute x equals infinity and get infinity over 2,"},{"Start":"15:15.805 ","End":"15:18.325","Text":"which is just infinity there,"},{"Start":"15:18.325 ","End":"15:19.825","Text":"got it in 1 line."},{"Start":"15:19.825 ","End":"15:25.445","Text":"I think we\u0027ll go for 1 more example of infinity over infinity. How about a set e^x?"},{"Start":"15:25.445 ","End":"15:32.360","Text":"Let\u0027s just take the limit as x goes to infinity of"},{"Start":"15:32.360 ","End":"15:39.125","Text":"the natural log of x and let\u0027s make it simple and just have it over x,"},{"Start":"15:39.125 ","End":"15:47.595","Text":"I once again can use these formulae to say that the natural log of infinity is infinity."},{"Start":"15:47.595 ","End":"15:50.820","Text":"We get that this is equal,"},{"Start":"15:50.820 ","End":"15:53.780","Text":"this is an infinity over infinity case,"},{"Start":"15:53.780 ","End":"15:58.190","Text":"and by L\u0027Hopital we just derive the top,"},{"Start":"15:58.190 ","End":"16:01.560","Text":"which is 1 over x."},{"Start":"16:08.760 ","End":"16:10.000","Text":"Yeah,"},{"Start":"16:10.000 ","End":"16:12.590","Text":"we just have to repeat"},{"Start":"16:12.590 ","End":"16:21.350","Text":"the limit as x goes to infinity and then derive top from bottom,"},{"Start":"16:21.350 ","End":"16:26.450","Text":"so yes, 1 over x and the derivative of x is"},{"Start":"16:26.450 ","End":"16:34.060","Text":"1 and so this thing is just 1 over x."},{"Start":"16:34.060 ","End":"16:37.880","Text":"We just get by substituting infinity,"},{"Start":"16:37.880 ","End":"16:44.699","Text":"1 over infinity, and 1 over infinity is equal to 0."},{"Start":"16:44.699 ","End":"16:52.310","Text":"That\u0027s it for this little exercise and we\u0027re done for part 1 of 4 parts in"},{"Start":"16:52.310 ","End":"17:00.330","Text":"all on the theory of behind L\u0027Hopital\u0027s rule. Till next time."}],"ID":1430},{"Watched":false,"Name":"Exercises 1-3","Duration":"2m 56s","ChapterTopicVideoID":8292,"CourseChapterTopicPlaylistID":52079,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8292.jpeg","UploadDate":"2017-03-02T01:05:27.3670000","DurationForVideoObject":"PT2M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.465","Text":"In this exercise we have to find the limit as given."},{"Start":"00:03.465 ","End":"00:08.220","Text":"We have here x squared minus x minus 6 over x squared minus 9,"},{"Start":"00:08.220 ","End":"00:09.435","Text":"now what\u0027s the problem?"},{"Start":"00:09.435 ","End":"00:13.065","Text":"The problem is that if we substitute x equals 3,"},{"Start":"00:13.065 ","End":"00:16.275","Text":"in here we get 3 squared minus 3 minus 6 is 0,"},{"Start":"00:16.275 ","End":"00:21.840","Text":"3 squared minus 9 is also 0 and so which 1 of those 0 over 0 cases?"},{"Start":"00:21.840 ","End":"00:23.670","Text":"What do we do with 0 over 0?"},{"Start":"00:23.670 ","End":"00:27.905","Text":"Well, there\u0027s many techniques but here we\u0027re going to use L\u0027Hopital system."},{"Start":"00:27.905 ","End":"00:31.355","Text":"L\u0027hopital\u0027s said that when we have a 0 over a 0"},{"Start":"00:31.355 ","End":"00:35.120","Text":"we can differentiate the numerator and denominator and we get"},{"Start":"00:35.120 ","End":"00:38.615","Text":"a completely different limit but the answer to this limit will be"},{"Start":"00:38.615 ","End":"00:43.015","Text":"exactly the same as the answer to this so differentiating this we get this,"},{"Start":"00:43.015 ","End":"00:45.075","Text":"differentiating this we get this."},{"Start":"00:45.075 ","End":"00:49.775","Text":"Here there\u0027s no problem to substitute x equals 3 twice 3 minus 1 is 5,"},{"Start":"00:49.775 ","End":"00:56.780","Text":"twice 3 is 6 and so we get 5 over 6 and that\u0027s our answer to this part."},{"Start":"00:56.780 ","End":"01:00.020","Text":"The next exercise we have is this 1."},{"Start":"01:00.020 ","End":"01:03.740","Text":"Once again, we\u0027re going to substitute x equals minus 5,"},{"Start":"01:03.740 ","End":"01:08.855","Text":"and if we substitute minus 5 we\u0027ll get 0 here and if you check it you will get 0 here."},{"Start":"01:08.855 ","End":"01:10.030","Text":"This is 1 of those"},{"Start":"01:10.030 ","End":"01:13.450","Text":"0 over 0 L\u0027hopital cases and"},{"Start":"01:13.450 ","End":"01:17.000","Text":"in which case what we do is we\u0027ll set of computing this limit,"},{"Start":"01:17.000 ","End":"01:21.395","Text":"we compute a different limit which is the 1 you obtain when you differentiate"},{"Start":"01:21.395 ","End":"01:23.810","Text":"the numerator separately and you differentiate"},{"Start":"01:23.810 ","End":"01:26.915","Text":"the denominator so 0 to x squared minus 50 gives us 4x."},{"Start":"01:26.915 ","End":"01:30.710","Text":"Here we get to the point where we can just substitute minus 5 and"},{"Start":"01:30.710 ","End":"01:34.759","Text":"like 4 times minus 5 is minus 20 plus 3 minus 17."},{"Start":"01:34.759 ","End":"01:37.910","Text":"In any event we eventually get this and we could have"},{"Start":"01:37.910 ","End":"01:41.765","Text":"dropped the minuses from here and what do we embed it for 20 over 17."},{"Start":"01:41.765 ","End":"01:46.880","Text":"Next exercise is this limit and an already I can guess that it\u0027s going to be"},{"Start":"01:46.880 ","End":"01:52.250","Text":"0 over 0 L\u0027hopital and if we put x equals 4 here,"},{"Start":"01:52.250 ","End":"01:53.870","Text":"twice 4 plus 1 is 9,"},{"Start":"01:53.870 ","End":"01:56.480","Text":"so that\u0027s 3, 4 plus 9 again is 9 minus 3,"},{"Start":"01:56.480 ","End":"02:00.950","Text":"3 minus 3 over 4 minus 4 it\u0027s 0 over 0 L\u0027hopital."},{"Start":"02:00.950 ","End":"02:05.165","Text":"We want to differentiate the numerator and differentiate the denominator."},{"Start":"02:05.165 ","End":"02:07.910","Text":"But before we differentiate derivative of square root is"},{"Start":"02:07.910 ","End":"02:11.780","Text":"1 over twice the square root of whatever it is but you have to multiply by"},{"Start":"02:11.780 ","End":"02:16.140","Text":"the internal derivative and so in our case what we get is for"},{"Start":"02:16.140 ","End":"02:21.035","Text":"the 2x plus 1 we get this and here\u0027s the internal derivative of 2x plus 1."},{"Start":"02:21.035 ","End":"02:23.375","Text":"Again, 1 over twice whatever it is,"},{"Start":"02:23.375 ","End":"02:28.325","Text":"internal derivative and the x minus 4 derived gives us just 1."},{"Start":"02:28.325 ","End":"02:31.790","Text":"Let\u0027s see what happens when we substitute x equals 4,"},{"Start":"02:31.790 ","End":"02:33.740","Text":"because at this point it\u0027s written as a mess and there\u0027s"},{"Start":"02:33.740 ","End":"02:35.750","Text":"no need to tidy it up because easiest just"},{"Start":"02:35.750 ","End":"02:40.400","Text":"to substitute so if we put x equals 4 this thing is 3 square root of 9."},{"Start":"02:40.400 ","End":"02:41.600","Text":"Anyway, it\u0027s all written here,"},{"Start":"02:41.600 ","End":"02:45.710","Text":"4 gives us square root of 9 and if we compute this the"},{"Start":"02:45.710 ","End":"02:48.500","Text":"over 1 disappears and the 2 cancels with the 2"},{"Start":"02:48.500 ","End":"02:51.320","Text":"so we get 1 over the square root of 9 which is the third,"},{"Start":"02:51.320 ","End":"02:57.210","Text":"and here square root of 9 is 3 so it\u0027s 1 over 6 and that\u0027s just a sixth."}],"ID":8463},{"Watched":false,"Name":"Exercises 4-5","Duration":"3m 13s","ChapterTopicVideoID":8293,"CourseChapterTopicPlaylistID":52079,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8293.jpeg","UploadDate":"2017-03-02T01:05:30.3100000","DurationForVideoObject":"PT3M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.225","Text":"Find this limit. If we put x equals 3,"},{"Start":"00:03.225 ","End":"00:05.370","Text":"I\u0027m betting it\u0027s going to be a 0 over 0,"},{"Start":"00:05.370 ","End":"00:08.820","Text":"3 squared is 9 plus 7 is 16,"},{"Start":"00:08.820 ","End":"00:11.340","Text":"whose square root is 4, 4 minus 4 is 0."},{"Start":"00:11.340 ","End":"00:12.900","Text":"So we have 0 over there,"},{"Start":"00:12.900 ","End":"00:14.250","Text":"3 minus 2 is 1,"},{"Start":"00:14.250 ","End":"00:15.590","Text":"square root is also 1."},{"Start":"00:15.590 ","End":"00:19.890","Text":"So 0 over 0 and we\u0027re going to do our L\u0027Hopital\u0027s rule."},{"Start":"00:19.890 ","End":"00:21.015","Text":"If we have 1 of those limits,"},{"Start":"00:21.015 ","End":"00:23.130","Text":"which is of the form 0 over 0."},{"Start":"00:23.130 ","End":"00:24.795","Text":"We can figure out a different limit,"},{"Start":"00:24.795 ","End":"00:27.960","Text":"which is derivative of the numerator over the derivative of"},{"Start":"00:27.960 ","End":"00:32.160","Text":"the denominator and that new limit will have the same answer as this one."},{"Start":"00:32.160 ","End":"00:35.730","Text":"Again, before we differentiate the square root formula appears"},{"Start":"00:35.730 ","End":"00:39.150","Text":"often, I have reminded you of it before and here it is again, square root."},{"Start":"00:39.150 ","End":"00:41.310","Text":"It gives you 1 over twice the square root."},{"Start":"00:41.310 ","End":"00:44.390","Text":"But if it\u0027s not x, you have to multiply by the derivative."},{"Start":"00:44.390 ","End":"00:45.515","Text":"If we do that,"},{"Start":"00:45.515 ","End":"00:47.225","Text":"then for the numerator,"},{"Start":"00:47.225 ","End":"00:51.200","Text":"we get this whole mass and further denominator, this mass."},{"Start":"00:51.200 ","End":"00:52.715","Text":"After we\u0027ve done that,"},{"Start":"00:52.715 ","End":"00:54.950","Text":"now we have the low petal version of the limit."},{"Start":"00:54.950 ","End":"00:58.640","Text":"Here I don\u0027t see any problem with putting x equals 3."},{"Start":"00:58.640 ","End":"01:00.095","Text":"We\u0027re not dividing by 0."},{"Start":"01:00.095 ","End":"01:02.600","Text":"So if we put x equals 3, like here,"},{"Start":"01:02.600 ","End":"01:07.355","Text":"we\u0027d have 3 squared plus 7 is 16, square root is 4."},{"Start":"01:07.355 ","End":"01:10.660","Text":"Yeah, we do get 1 over square root of 16 times the"},{"Start":"01:10.660 ","End":"01:14.030","Text":"3 and the 2s cancel and so on with the denominator."},{"Start":"01:14.030 ","End":"01:17.450","Text":"All that remains is to just say that the square root of 16 is"},{"Start":"01:17.450 ","End":"01:21.770","Text":"4 and once we get 3 over 4 divided by 1/2,"},{"Start":"01:21.770 ","End":"01:24.475","Text":"is 3 over 4 times 2 over 1."},{"Start":"01:24.475 ","End":"01:26.765","Text":"Anyway, this is what it comes out to be."},{"Start":"01:26.765 ","End":"01:28.850","Text":"Let\u0027s continue to the next 1,"},{"Start":"01:28.850 ","End":"01:31.460","Text":"and this is the next exercise."},{"Start":"01:31.460 ","End":"01:35.630","Text":"We have this time a cube root minus a square root."},{"Start":"01:35.630 ","End":"01:37.130","Text":"If we put x equals 1,"},{"Start":"01:37.130 ","End":"01:40.115","Text":"0 here, put x equals 1, then,"},{"Start":"01:40.115 ","End":"01:42.620","Text":"then we get 2 minus 1 is 1,"},{"Start":"01:42.620 ","End":"01:46.220","Text":"whose cube root is also 1 and here\u0027s 1, we get 0 over 0."},{"Start":"01:46.220 ","End":"01:48.215","Text":"We\u0027d better use L\u0027Hopital\u0027s rule,"},{"Start":"01:48.215 ","End":"01:50.810","Text":"which is to say differentiate the numerator,"},{"Start":"01:50.810 ","End":"01:53.030","Text":"differentiate the denominator and the answer to"},{"Start":"01:53.030 ","End":"01:56.570","Text":"this new exercise will be the same as the answer to the old exercise."},{"Start":"01:56.570 ","End":"01:59.840","Text":"We\u0027re using the fact that the cube root of something is to"},{"Start":"01:59.840 ","End":"02:03.410","Text":"the power of the third and the square root is to the power of 1/2."},{"Start":"02:03.410 ","End":"02:07.055","Text":"We have to differentiate this using all the various tricks."},{"Start":"02:07.055 ","End":"02:09.710","Text":"One of the things is that a third is nothing special as far"},{"Start":"02:09.710 ","End":"02:12.845","Text":"as the power rule goes at something to the n,"},{"Start":"02:12.845 ","End":"02:14.960","Text":"you multiply by the 1/3,"},{"Start":"02:14.960 ","End":"02:16.850","Text":"you reduce the power by 1."},{"Start":"02:16.850 ","End":"02:19.440","Text":"So, from 1/3 it goes down to minus 2/3."},{"Start":"02:19.440 ","End":"02:22.190","Text":"Whenever it was, in this case 2x squared minus 1,"},{"Start":"02:22.190 ","End":"02:24.260","Text":"you also have to take the inner derivative."},{"Start":"02:24.260 ","End":"02:25.565","Text":"If we do that,"},{"Start":"02:25.565 ","End":"02:30.320","Text":"what we have here is here we have the 1/3 box to the minus 2/3."},{"Start":"02:30.320 ","End":"02:33.470","Text":"Here we have the minus 1/2, whatever it is,"},{"Start":"02:33.470 ","End":"02:37.355","Text":"box to the power of minus 1/2, we subtracted 1,"},{"Start":"02:37.355 ","End":"02:41.840","Text":"the 4x is from the 2x squared minus 1 from the x it\u0027s just 1."},{"Start":"02:41.840 ","End":"02:45.440","Text":"This is what we get, denominator of course gives us just 1,"},{"Start":"02:45.440 ","End":"02:49.760","Text":"and now we just can substitute x equals 1 shouldn\u0027t be any problem."},{"Start":"02:49.760 ","End":"02:51.980","Text":"If we substitute x equals 1, the 1 in"},{"Start":"02:51.980 ","End":"02:55.955","Text":"the round brackets is the 1 which we have substituted here."},{"Start":"02:55.955 ","End":"02:57.260","Text":"We dropped off all the 1s."},{"Start":"02:57.260 ","End":"02:59.195","Text":"We have 1/3 times 4,"},{"Start":"02:59.195 ","End":"03:01.415","Text":"which is 4/3 minus 1/2."},{"Start":"03:01.415 ","End":"03:02.540","Text":"If you do your fractions,"},{"Start":"03:02.540 ","End":"03:04.205","Text":"4/3 minus 1/2,"},{"Start":"03:04.205 ","End":"03:06.485","Text":"you could put it in terms of 6th,"},{"Start":"03:06.485 ","End":"03:09.010","Text":"4/3 is 8/6, 1/2 is 3/6."},{"Start":"03:09.010 ","End":"03:14.430","Text":"So we end up with only 5/6 and that\u0027s the answer to that 1."}],"ID":8464},{"Watched":false,"Name":"Exercises 6-7","Duration":"1m 47s","ChapterTopicVideoID":8294,"CourseChapterTopicPlaylistID":52079,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8294.jpeg","UploadDate":"2017-03-02T01:05:31.6530000","DurationForVideoObject":"PT1M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.780","Text":"We have the limit as x goes to 0 of this thing."},{"Start":"00:03.780 ","End":"00:05.070","Text":"Now, if we substitute,"},{"Start":"00:05.070 ","End":"00:08.250","Text":"we see that the denominator is 0,"},{"Start":"00:08.250 ","End":"00:11.475","Text":"and so is the numerator because e^0 is 1."},{"Start":"00:11.475 ","End":"00:15.330","Text":"What we have, it\u0027s we\u0027re getting a 0 over 0 situation."},{"Start":"00:15.330 ","End":"00:17.700","Text":"By L\u0027Hopital\u0027s rule, it\u0027s equal to"},{"Start":"00:17.700 ","End":"00:20.310","Text":"the limit of the derivative here and the derivative here,"},{"Start":"00:20.310 ","End":"00:22.485","Text":"that\u0027s just Alpha is name."},{"Start":"00:22.485 ","End":"00:25.815","Text":"We differentiate the numerator e^x is just e^x."},{"Start":"00:25.815 ","End":"00:27.885","Text":"The constant goes to 0,"},{"Start":"00:27.885 ","End":"00:30.945","Text":"e^x goes to 1, so this is just e^x."},{"Start":"00:30.945 ","End":"00:32.730","Text":"Here we can substitute x equals 0,"},{"Start":"00:32.730 ","End":"00:33.870","Text":"so we get e^0."},{"Start":"00:33.870 ","End":"00:36.285","Text":"What is e^0? It\u0027s just 1."},{"Start":"00:36.285 ","End":"00:43.085","Text":"Here\u0027s an unusual 1 the limit as x goes to 0 of a^x minus b^x over x."},{"Start":"00:43.085 ","End":"00:44.360","Text":"In order for this to make sense,"},{"Start":"00:44.360 ","End":"00:47.435","Text":"we need to require that a and b both be positive,"},{"Start":"00:47.435 ","End":"00:52.445","Text":"then the exponent is defined so just rewriting it here for convenience."},{"Start":"00:52.445 ","End":"00:55.520","Text":"What we get is that a^0 is 1,"},{"Start":"00:55.520 ","End":"00:58.350","Text":"b^0 is 1 and 0 is 0."},{"Start":"00:58.350 ","End":"01:01.605","Text":"Again, we\u0027re in a situation of 0 over 0."},{"Start":"01:01.605 ","End":"01:05.255","Text":"This limit equals by L\u0027Hopital a different limit,"},{"Start":"01:05.255 ","End":"01:09.830","Text":"which is obtained by differentiating separately the numerator and the denominator."},{"Start":"01:09.830 ","End":"01:15.170","Text":"Now the denominator is easy to see is 1 and there is a formula that a^x is"},{"Start":"01:15.170 ","End":"01:19.085","Text":"e^x natural log of a and what we have"},{"Start":"01:19.085 ","End":"01:23.870","Text":"here is now at a point where there is no problem in substituting x equals 0."},{"Start":"01:23.870 ","End":"01:27.180","Text":"If we do it, we just get this 0 it gives us"},{"Start":"01:27.180 ","End":"01:31.065","Text":"1 natural log of a and this also gives us e^0, 1."},{"Start":"01:31.065 ","End":"01:34.595","Text":"We just get natural log of a minus natural log of b."},{"Start":"01:34.595 ","End":"01:37.145","Text":"For those who like to use logarithms,"},{"Start":"01:37.145 ","End":"01:38.705","Text":"I could have left it like this,"},{"Start":"01:38.705 ","End":"01:41.990","Text":"but the logarithm of a quotient is the difference of the logs and"},{"Start":"01:41.990 ","End":"01:45.560","Text":"that works backwards as well so we get natural log of a."},{"Start":"01:45.560 ","End":"01:48.480","Text":"Yes, so that\u0027s it for this exercise."}],"ID":8465},{"Watched":false,"Name":"Exercises 8-11","Duration":"6m 15s","ChapterTopicVideoID":8295,"CourseChapterTopicPlaylistID":52079,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8295.jpeg","UploadDate":"2017-03-02T01:05:37.0300000","DurationForVideoObject":"PT6M15S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.625","Text":"This expression as x goes to 0,"},{"Start":"00:02.625 ","End":"00:04.065","Text":"why don\u0027t we just substitute?"},{"Start":"00:04.065 ","End":"00:07.605","Text":"Well, we can\u0027t because this is 0 and e^0 is 1."},{"Start":"00:07.605 ","End":"00:10.530","Text":"1 minus 0 minus 1 is also 0."},{"Start":"00:10.530 ","End":"00:12.410","Text":"I\u0027m rewriting it here for convenience."},{"Start":"00:12.410 ","End":"00:15.435","Text":"What we have here is a 0 over 0 situation,"},{"Start":"00:15.435 ","End":"00:17.745","Text":"which means that we use L\u0027Hopital\u0027s rule,"},{"Start":"00:17.745 ","End":"00:19.470","Text":"which says, \"Don\u0027t compute this limit,"},{"Start":"00:19.470 ","End":"00:24.000","Text":"compute a different one which is obtained by differentiating top and bottom.\""},{"Start":"00:24.000 ","End":"00:26.655","Text":"Top gives us e^x minus 1,"},{"Start":"00:26.655 ","End":"00:29.280","Text":"the bottom gives us from x squared 2x."},{"Start":"00:29.280 ","End":"00:33.630","Text":"Here, we still can\u0027t substitute x equals 0 because again,"},{"Start":"00:33.630 ","End":"00:36.015","Text":"we\u0027ve got a 0 over 0 situation."},{"Start":"00:36.015 ","End":"00:38.985","Text":"However, we can use L\u0027Hopital\u0027s rule again."},{"Start":"00:38.985 ","End":"00:41.480","Text":"Differentiating the numerator, we get e^x."},{"Start":"00:41.480 ","End":"00:43.430","Text":"The denominator just gives us 2."},{"Start":"00:43.430 ","End":"00:46.460","Text":"Here, there\u0027s no reason not to substitute x equals 0."},{"Start":"00:46.460 ","End":"00:49.415","Text":"E^0 is 1, and that just gives us 1.5."},{"Start":"00:49.415 ","End":"00:50.915","Text":"That\u0027s it for this one."},{"Start":"00:50.915 ","End":"00:52.250","Text":"Let\u0027s see if there\u0027s anymore."},{"Start":"00:52.250 ","End":"00:56.060","Text":"Here\u0027s 1, put x equals 0 and we get 2 minus,"},{"Start":"00:56.060 ","End":"00:57.890","Text":"so these 2 middle terms are 0."},{"Start":"00:57.890 ","End":"00:59.690","Text":"We get twice e^0,"},{"Start":"00:59.690 ","End":"01:02.240","Text":"which is twice 1 minus 2x 0,"},{"Start":"01:02.240 ","End":"01:04.895","Text":"and then the denominator, if x is 0, it\u0027s 0."},{"Start":"01:04.895 ","End":"01:06.940","Text":"Pull out L\u0027Hopital\u0027s rule,"},{"Start":"01:06.940 ","End":"01:10.670","Text":"and what we get is a different limit by differentiating top and bottom."},{"Start":"01:10.670 ","End":"01:14.425","Text":"Here, 2e^x minus 2x from there."},{"Start":"01:14.425 ","End":"01:16.225","Text":"The 2x gives us minus 2."},{"Start":"01:16.225 ","End":"01:19.840","Text":"On the bottom, we get 3 times 2 is 6 and lower the power by 1."},{"Start":"01:19.840 ","End":"01:21.515","Text":"Let\u0027s see, are we okay now,"},{"Start":"01:21.515 ","End":"01:23.405","Text":"we put x equals 0."},{"Start":"01:23.405 ","End":"01:28.820","Text":"Again, we get 0 in the bottom and then in the top we get 2 minus 2. You\u0027ve got the idea."},{"Start":"01:28.820 ","End":"01:31.895","Text":"We try again with L\u0027Hopital, and this time,"},{"Start":"01:31.895 ","End":"01:35.105","Text":"derivative is 2e^x minus 2 on the top,"},{"Start":"01:35.105 ","End":"01:36.765","Text":"on the bottom 12x,"},{"Start":"01:36.765 ","End":"01:39.240","Text":"put x equals 0, again, 0 over 0."},{"Start":"01:39.240 ","End":"01:41.185","Text":"I will do it once more and see,"},{"Start":"01:41.185 ","End":"01:42.930","Text":"finally something we can work with."},{"Start":"01:42.930 ","End":"01:47.000","Text":"Here we can simply substitute x equals 0 because there\u0027s no problem on the bottom,"},{"Start":"01:47.000 ","End":"01:48.365","Text":"no problem on the top."},{"Start":"01:48.365 ","End":"01:50.750","Text":"We do get actually 2 over 12."},{"Start":"01:50.750 ","End":"01:52.370","Text":"I didn\u0027t write the 2 over 12,"},{"Start":"01:52.370 ","End":"01:55.025","Text":"I straight away canceled it to be 1/6,"},{"Start":"01:55.025 ","End":"01:56.855","Text":"which is the answer to this one."},{"Start":"01:56.855 ","End":"02:00.215","Text":"Moving on, there\u0027s an abundance of exercises."},{"Start":"02:00.215 ","End":"02:05.235","Text":"For convenience, again, repeated the exercise in a larger font in the solution."},{"Start":"02:05.235 ","End":"02:07.390","Text":"Let\u0027s see if we tried to substitute."},{"Start":"02:07.390 ","End":"02:11.135","Text":"Put x equals 1, 1 squared plus 1 minus 2 is 0."},{"Start":"02:11.135 ","End":"02:12.920","Text":"Natural log of 1 is 0,"},{"Start":"02:12.920 ","End":"02:15.005","Text":"0 minus 1 plus 1,"},{"Start":"02:15.005 ","End":"02:16.565","Text":"now it\u0027s 0 over 0."},{"Start":"02:16.565 ","End":"02:21.035","Text":"0 over 0, bring out your L\u0027Hopital, the 0, 0."},{"Start":"02:21.035 ","End":"02:24.005","Text":"Derivative here is 1 over x,"},{"Start":"02:24.005 ","End":"02:26.435","Text":"derivative here is minus 1 and nothing."},{"Start":"02:26.435 ","End":"02:28.790","Text":"Here, 2x minus 2 and nothing."},{"Start":"02:28.790 ","End":"02:30.200","Text":"This is what we have here."},{"Start":"02:30.200 ","End":"02:32.000","Text":"Again, putting x equals 1,"},{"Start":"02:32.000 ","End":"02:35.270","Text":"we get 1 minus 1 over 2 minus 2, which is 0 over 0."},{"Start":"02:35.270 ","End":"02:37.715","Text":"How about trying it yet again,"},{"Start":"02:37.715 ","End":"02:39.950","Text":"L\u0027Hopital for 0 over 0."},{"Start":"02:39.950 ","End":"02:41.960","Text":"Now there\u0027s no problem to put x equals 1."},{"Start":"02:41.960 ","End":"02:44.420","Text":"Put x equals 1 here, that\u0027s 1 over 1,"},{"Start":"02:44.420 ","End":"02:46.340","Text":"that\u0027s just minus 1 over 2,"},{"Start":"02:46.340 ","End":"02:47.915","Text":"and that\u0027s all there is."},{"Start":"02:47.915 ","End":"02:50.060","Text":"The exercise is this,"},{"Start":"02:50.060 ","End":"02:53.450","Text":"but we just didn\u0027t write it out again in the solution part."},{"Start":"02:53.450 ","End":"02:57.335","Text":"If we look at this and we put x equals infinity,"},{"Start":"02:57.335 ","End":"02:58.430","Text":"what we get is,"},{"Start":"02:58.430 ","End":"03:00.185","Text":"using the usual techniques,"},{"Start":"03:00.185 ","End":"03:02.420","Text":"divide top and bottom by x squared,"},{"Start":"03:02.420 ","End":"03:04.670","Text":"then we get 1 plus 1 over x squared,"},{"Start":"03:04.670 ","End":"03:06.580","Text":"1 minus 1 over x squared."},{"Start":"03:06.580 ","End":"03:08.000","Text":"When x goes to infinity,"},{"Start":"03:08.000 ","End":"03:09.770","Text":"the 1 over x squared goes to 0."},{"Start":"03:09.770 ","End":"03:12.155","Text":"So we get natural log of 1 over 1,"},{"Start":"03:12.155 ","End":"03:14.165","Text":"natural log of 1 is 0."},{"Start":"03:14.165 ","End":"03:15.440","Text":"Likewise, as we said,"},{"Start":"03:15.440 ","End":"03:18.275","Text":"1 over x squared is 0 when x goes to infinity."},{"Start":"03:18.275 ","End":"03:20.270","Text":"We do have a 0 over 0 here."},{"Start":"03:20.270 ","End":"03:22.250","Text":"Now if you need to do that more slowly,"},{"Start":"03:22.250 ","End":"03:23.915","Text":"you should to it on your own on the side."},{"Start":"03:23.915 ","End":"03:27.300","Text":"What we do is we\u0027re going to differentiate the top and the bottom."},{"Start":"03:27.300 ","End":"03:31.045","Text":"Natural log of something when you derive it is just 1 over that something,"},{"Start":"03:31.045 ","End":"03:33.420","Text":"just like natural log of x goes to 1 over x."},{"Start":"03:33.420 ","End":"03:35.810","Text":"Because it\u0027s a something and not just an x,"},{"Start":"03:35.810 ","End":"03:39.170","Text":"we have to also multiply by the derivative of that something."},{"Start":"03:39.170 ","End":"03:42.365","Text":"Using this formula and applying it to L\u0027Hopital,"},{"Start":"03:42.365 ","End":"03:44.345","Text":"meaning we want to differentiate the top"},{"Start":"03:44.345 ","End":"03:47.130","Text":"separately and we want to differentiate the bottom separately,"},{"Start":"03:47.130 ","End":"03:48.800","Text":"what we get is a mass,"},{"Start":"03:48.800 ","End":"03:52.530","Text":"and this mass is 1 over what was in the numerator."},{"Start":"03:52.530 ","End":"03:55.535","Text":"We just take 1 over that times the internal derivative,"},{"Start":"03:55.535 ","End":"03:56.660","Text":"haven\u0027t derived it yet,"},{"Start":"03:56.660 ","End":"03:58.790","Text":"but 1 over x squared is well known,"},{"Start":"03:58.790 ","End":"04:00.590","Text":"is minus 2 over x cubed."},{"Start":"04:00.590 ","End":"04:02.750","Text":"Basically, in your head you say x^ minus 2,"},{"Start":"04:02.750 ","End":"04:04.835","Text":"so it\u0027s minus 2x^ minus 3."},{"Start":"04:04.835 ","End":"04:08.870","Text":"Anyway, this is what we get and certainly need some simplification."},{"Start":"04:08.870 ","End":"04:11.945","Text":"What it comes out to be is that if we do"},{"Start":"04:11.945 ","End":"04:15.575","Text":"quite a bit of algebra and we do this thing using the quotient rule,"},{"Start":"04:15.575 ","End":"04:20.030","Text":"1 over this, we just invert because 1 over a fraction is the inverted fraction."},{"Start":"04:20.030 ","End":"04:21.290","Text":"This stays where it is,"},{"Start":"04:21.290 ","End":"04:23.975","Text":"and derivative of this we do with the quotient rule,"},{"Start":"04:23.975 ","End":"04:27.110","Text":"which is the derivative of the top times the bottom minus"},{"Start":"04:27.110 ","End":"04:30.340","Text":"the top times the derivative of the bottom over the bottom squared."},{"Start":"04:30.340 ","End":"04:34.300","Text":"Then continuing simplification, we get this x squared"},{"Start":"04:34.300 ","End":"04:39.025","Text":"minus 1 here cancels with 1 of these x squared minus 1s."},{"Start":"04:39.025 ","End":"04:42.290","Text":"This part leaves us with 1 over x squared plus 1."},{"Start":"04:42.290 ","End":"04:44.285","Text":"Here, this is the x squared minus 1."},{"Start":"04:44.285 ","End":"04:45.635","Text":"But if we open this up,"},{"Start":"04:45.635 ","End":"04:47.690","Text":"we get 2x cubed here,"},{"Start":"04:47.690 ","End":"04:50.105","Text":"minus 2x cubed, which cancels out."},{"Start":"04:50.105 ","End":"04:51.860","Text":"From here, minus 2x,"},{"Start":"04:51.860 ","End":"04:53.420","Text":"and from here, minus 2x."},{"Start":"04:53.420 ","End":"04:56.520","Text":"This whole top leaves us with the minus 4x."},{"Start":"04:56.520 ","End":"04:57.900","Text":"That\u0027s what we\u0027re left with,"},{"Start":"04:57.900 ","End":"05:00.050","Text":"and then denominator is the same."},{"Start":"05:00.050 ","End":"05:02.345","Text":"The next step brings us to this,"},{"Start":"05:02.345 ","End":"05:07.630","Text":"which we got by taking the x squared plus 1 and the x squared minus 1 to the bottom."},{"Start":"05:07.630 ","End":"05:10.055","Text":"Then when you divide by a fraction,"},{"Start":"05:10.055 ","End":"05:12.440","Text":"you multiply by the inverse fraction,"},{"Start":"05:12.440 ","End":"05:15.380","Text":"which is x cubed over 2 with a minus."},{"Start":"05:15.380 ","End":"05:16.685","Text":"The minus goes with the minus."},{"Start":"05:16.685 ","End":"05:21.320","Text":"If we take the minus 4x and multiply it by x cubed over 2,"},{"Start":"05:21.320 ","End":"05:22.400","Text":"the reverse of this,"},{"Start":"05:22.400 ","End":"05:27.080","Text":"so the x times the x cubed becomes x^4 and the 4 over minus 2,"},{"Start":"05:27.080 ","End":"05:29.035","Text":"or the 4 over 2 stays as 2."},{"Start":"05:29.035 ","End":"05:31.010","Text":"This is what we get here."},{"Start":"05:31.010 ","End":"05:35.555","Text":"Then proceeding, what we get is if you take this thing,"},{"Start":"05:35.555 ","End":"05:37.550","Text":"it\u0027s actually difference of squares."},{"Start":"05:37.550 ","End":"05:40.865","Text":"It really should be x^4 minus 1,"},{"Start":"05:40.865 ","End":"05:41.990","Text":"because this is a plus b a"},{"Start":"05:41.990 ","End":"05:44.780","Text":"minus b, so it should be a squared minus b squared,"},{"Start":"05:44.780 ","End":"05:46.640","Text":"that there should be a minus 1 here."},{"Start":"05:46.640 ","End":"05:48.680","Text":"But we\u0027ve skipped a step."},{"Start":"05:48.680 ","End":"05:51.035","Text":"When you take a polynomial over a polynomial,"},{"Start":"05:51.035 ","End":"05:52.220","Text":"it\u0027s only the leading,"},{"Start":"05:52.220 ","End":"05:54.875","Text":"the highest powers that determine what the limit is."},{"Start":"05:54.875 ","End":"05:57.230","Text":"So we can ignore that minus 1 that was here."},{"Start":"05:57.230 ","End":"06:01.280","Text":"Or we can just afterwards divide by x^4 over x^4,"},{"Start":"06:01.280 ","End":"06:05.585","Text":"top and bottom and you\u0027d get 2 over 1 minus 1 over x^4."},{"Start":"06:05.585 ","End":"06:08.285","Text":"Anyway, the other 1 here is not consequential."},{"Start":"06:08.285 ","End":"06:13.025","Text":"That leaves us with just the quotient of the leading coefficients, which is 2."},{"Start":"06:13.025 ","End":"06:16.680","Text":"That in fact is the answer to this one."}],"ID":8466},{"Watched":false,"Name":"Exercises 12-14","Duration":"4m 22s","ChapterTopicVideoID":8296,"CourseChapterTopicPlaylistID":52079,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8296.jpeg","UploadDate":"2017-03-02T01:05:40.7930000","DurationForVideoObject":"PT4M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.770","Text":"Here is the next exercise."},{"Start":"00:01.770 ","End":"00:04.950","Text":"This time its limit x goes to 0 of this thing."},{"Start":"00:04.950 ","End":"00:09.180","Text":"First thing we try is to substitute and see if anything goes wrong with that."},{"Start":"00:09.180 ","End":"00:11.745","Text":"Well, yes, of course the denominator is 0,"},{"Start":"00:11.745 ","End":"00:13.665","Text":"but if we try the numerator also,"},{"Start":"00:13.665 ","End":"00:14.970","Text":"0 plus 1 is 1."},{"Start":"00:14.970 ","End":"00:16.995","Text":"Natural log of 1 is 0,"},{"Start":"00:16.995 ","End":"00:19.155","Text":"squared is still 0 plus 0."},{"Start":"00:19.155 ","End":"00:21.870","Text":"We have a 0 over 0 L\u0027Hopital."},{"Start":"00:21.870 ","End":"00:25.950","Text":"Also, this notation is misleading that the 2 there,"},{"Start":"00:25.950 ","End":"00:28.410","Text":"it means natural log of x plus 1,"},{"Start":"00:28.410 ","End":"00:29.700","Text":"the whole thing squared,"},{"Start":"00:29.700 ","End":"00:31.995","Text":"so it would be a better form to write it in."},{"Start":"00:31.995 ","End":"00:35.610","Text":"What we\u0027re going to do is use L\u0027Hopital to say that if we take"},{"Start":"00:35.610 ","End":"00:40.625","Text":"a different limit where we differentiate both numerator and denominator separately,"},{"Start":"00:40.625 ","End":"00:43.250","Text":"that will be the same answer as the original one."},{"Start":"00:43.250 ","End":"00:46.040","Text":"What we do is we need a formula,"},{"Start":"00:46.040 ","End":"00:47.810","Text":"first of all, for something squared."},{"Start":"00:47.810 ","End":"00:51.890","Text":"When it\u0027s derived just as the derivative of x squared is 2x,"},{"Start":"00:51.890 ","End":"00:53.375","Text":"when it\u0027s something else squared,"},{"Start":"00:53.375 ","End":"00:55.040","Text":"it\u0027s twice that something else,"},{"Start":"00:55.040 ","End":"00:56.960","Text":"but times the internal derivative,"},{"Start":"00:56.960 ","End":"00:58.250","Text":"the derivative of that thing."},{"Start":"00:58.250 ","End":"01:02.525","Text":"In our case we get twice this natural log of x plus 1,"},{"Start":"01:02.525 ","End":"01:04.175","Text":"but times the internal,"},{"Start":"01:04.175 ","End":"01:06.110","Text":"which is the derivative of what this is."},{"Start":"01:06.110 ","End":"01:07.640","Text":"We\u0027ll see what that is in a moment."},{"Start":"01:07.640 ","End":"01:10.100","Text":"Plus 1 from the x and 1 here."},{"Start":"01:10.100 ","End":"01:12.650","Text":"Now what we\u0027re left is differentiating this."},{"Start":"01:12.650 ","End":"01:16.160","Text":"What we\u0027re going to do is mention that this is a chain rule,"},{"Start":"01:16.160 ","End":"01:18.125","Text":"also the formula with the square."},{"Start":"01:18.125 ","End":"01:20.495","Text":"It\u0027s when you have a function of a function,"},{"Start":"01:20.495 ","End":"01:21.890","Text":"in this case the square function,"},{"Start":"01:21.890 ","End":"01:23.885","Text":"in this case, the natural log function,"},{"Start":"01:23.885 ","End":"01:27.260","Text":"that when you derive the natural log of something which is not x,"},{"Start":"01:27.260 ","End":"01:31.010","Text":"then it\u0027s 1 over that something but times the internal derivative."},{"Start":"01:31.010 ","End":"01:33.365","Text":"In this case, what it brings us to is"},{"Start":"01:33.365 ","End":"01:36.870","Text":"the 2 stays as 2 because multiplicative constants just stay."},{"Start":"01:36.870 ","End":"01:39.935","Text":"What we need is a derivative of natural log of x plus 1."},{"Start":"01:39.935 ","End":"01:46.130","Text":"Natural log gives us the 1 over x plus 1 and times the internal derivative,"},{"Start":"01:46.130 ","End":"01:48.005","Text":"which is the derivative of x plus 1,"},{"Start":"01:48.005 ","End":"01:49.565","Text":"which is just 1."},{"Start":"01:49.565 ","End":"01:53.015","Text":"So the derivative of this is just 1 over x plus 1,"},{"Start":"01:53.015 ","End":"01:54.785","Text":"and we didn\u0027t put the times 1."},{"Start":"01:54.785 ","End":"01:57.215","Text":"This plus 1 is just copying from here."},{"Start":"01:57.215 ","End":"02:01.355","Text":"Basically all we did is differentiated this and came out with 1 over x plus 1."},{"Start":"02:01.355 ","End":"02:04.550","Text":"Then there\u0027s just a bit of algebra to be done because it has"},{"Start":"02:04.550 ","End":"02:07.850","Text":"no reason why we can\u0027t substitute x equals 0 at this point,"},{"Start":"02:07.850 ","End":"02:09.920","Text":"so that\u0027s just 1, x is 0."},{"Start":"02:09.920 ","End":"02:14.435","Text":"Just remember, the natural log of 1 is 0 so this is twice 0."},{"Start":"02:14.435 ","End":"02:18.125","Text":"This is 1 over 0 plus 1 which is just 1,"},{"Start":"02:18.125 ","End":"02:19.760","Text":"and that\u0027s this one done."},{"Start":"02:19.760 ","End":"02:21.620","Text":"This time we have a tangent."},{"Start":"02:21.620 ","End":"02:24.080","Text":"Well, let\u0027s see what happens when we put x equals 0."},{"Start":"02:24.080 ","End":"02:27.060","Text":"The bottom is 0 and tangent of 0 is 0."},{"Start":"02:27.060 ","End":"02:28.280","Text":"Tangent is, after all,"},{"Start":"02:28.280 ","End":"02:31.460","Text":"sine over cosine and sine of 0 is certainly 0."},{"Start":"02:31.460 ","End":"02:36.110","Text":"What we need to do is take this expression and use L\u0027Hopital\u0027s rule,"},{"Start":"02:36.110 ","End":"02:38.345","Text":"which is to equate this to a different limit,"},{"Start":"02:38.345 ","End":"02:41.570","Text":"which is the one obtained by differentiating top and bottom."},{"Start":"02:41.570 ","End":"02:46.940","Text":"This 0 over 0 by L\u0027Hopital is equal to derivative of the bottom is just 1."},{"Start":"02:46.940 ","End":"02:48.500","Text":"If you look at your formula sheet,"},{"Start":"02:48.500 ","End":"02:51.155","Text":"you\u0027ll see that the tangent of x has a derivative,"},{"Start":"02:51.155 ","End":"02:53.615","Text":"which in some books is called 1 over cosine"},{"Start":"02:53.615 ","End":"02:56.990","Text":"squared x and in other books they say the secant squared of x,"},{"Start":"02:56.990 ","End":"02:58.280","Text":"which is 1 over cosine."},{"Start":"02:58.280 ","End":"02:59.510","Text":"In any event at this point,"},{"Start":"02:59.510 ","End":"03:01.805","Text":"we can just substitute x equals 0."},{"Start":"03:01.805 ","End":"03:04.310","Text":"Remember that cosine of 0 is 1."},{"Start":"03:04.310 ","End":"03:06.920","Text":"All we\u0027re left when we do this is end up with 1."},{"Start":"03:06.920 ","End":"03:08.740","Text":"Continuing to this expression,"},{"Start":"03:08.740 ","End":"03:12.650","Text":"we have to have b not equal to 0 because we don\u0027t want a b in the denominator."},{"Start":"03:12.650 ","End":"03:15.380","Text":"What we have here is the limit when b is not equal to"},{"Start":"03:15.380 ","End":"03:19.415","Text":"0 of sine of ax squared over bx squared."},{"Start":"03:19.415 ","End":"03:23.165","Text":"We know that the sine of 0 is 0."},{"Start":"03:23.165 ","End":"03:25.385","Text":"If x is 0, this whole thing is 0."},{"Start":"03:25.385 ","End":"03:26.870","Text":"We have sine of 0, which is 0,"},{"Start":"03:26.870 ","End":"03:29.330","Text":"and on the bottom, since x is 0 it\u0027s also 0."},{"Start":"03:29.330 ","End":"03:32.630","Text":"Here we have another case of 0 over 0,"},{"Start":"03:32.630 ","End":"03:34.700","Text":"so we\u0027ll whip out our L\u0027Hopital."},{"Start":"03:34.700 ","End":"03:38.270","Text":"This is the formula we\u0027re going to need because when we differentiate the numerator,"},{"Start":"03:38.270 ","End":"03:39.965","Text":"we\u0027re going to need the formula for"},{"Start":"03:39.965 ","End":"03:43.490","Text":"the chain rule with the sine being the external functions."},{"Start":"03:43.490 ","End":"03:45.590","Text":"The sine of something derived is the cosine"},{"Start":"03:45.590 ","End":"03:47.800","Text":"of that something times the internal derivative."},{"Start":"03:47.800 ","End":"03:50.675","Text":"In this case, the internal is the ax squared."},{"Start":"03:50.675 ","End":"03:52.880","Text":"What we get, we differentiate the top is"},{"Start":"03:52.880 ","End":"03:55.820","Text":"the cosine of ax squared times the internal derivative,"},{"Start":"03:55.820 ","End":"03:57.965","Text":"which is 2 times a times x."},{"Start":"03:57.965 ","End":"04:00.320","Text":"On the bottom, it\u0027s simply x squared is 2x,"},{"Start":"04:00.320 ","End":"04:02.405","Text":"so bx squared is 2bx."},{"Start":"04:02.405 ","End":"04:05.550","Text":"We can\u0027t quite substitute x equals 0 yet,"},{"Start":"04:05.550 ","End":"04:08.210","Text":"but if we cancel this x with this x,"},{"Start":"04:08.210 ","End":"04:12.349","Text":"then we could substitute x equals 0 because then if x is 0,"},{"Start":"04:12.349 ","End":"04:13.700","Text":"the 2 would cancel with the 2,"},{"Start":"04:13.700 ","End":"04:16.175","Text":"the x with the x, so at the bottom we have b."},{"Start":"04:16.175 ","End":"04:18.230","Text":"Here, cosine of 0 is 1,"},{"Start":"04:18.230 ","End":"04:19.730","Text":"so all we\u0027re left with is the a."},{"Start":"04:19.730 ","End":"04:23.490","Text":"So the actual answer is a over b."}],"ID":8467},{"Watched":false,"Name":"Exercises 15-17","Duration":"3m 36s","ChapterTopicVideoID":8282,"CourseChapterTopicPlaylistID":52079,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8282.jpeg","UploadDate":"2017-03-02T01:07:58.2800000","DurationForVideoObject":"PT3M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.030","Text":"Sine of ax over the sine of bx,"},{"Start":"00:03.030 ","End":"00:05.790","Text":"but we have to exclude the case of b is 0,"},{"Start":"00:05.790 ","End":"00:08.325","Text":"otherwise we\u0027ll have a 0 on the denominator."},{"Start":"00:08.325 ","End":"00:13.035","Text":"We can quickly see that it\u0027s a case of L\u0027Hôpital because when x is 0,"},{"Start":"00:13.035 ","End":"00:15.255","Text":"ax is 0 and sine of 0 is 0."},{"Start":"00:15.255 ","End":"00:16.770","Text":"Likewise, when x is 0,"},{"Start":"00:16.770 ","End":"00:17.970","Text":"sine bx is 0,"},{"Start":"00:17.970 ","End":"00:19.455","Text":"so it\u0027s 0 over 0."},{"Start":"00:19.455 ","End":"00:23.569","Text":"So we want to differentiate the top and the bottom for L\u0027Hôpital,"},{"Start":"00:23.569 ","End":"00:26.840","Text":"but we just need to remember this formula that we had not long ago,"},{"Start":"00:26.840 ","End":"00:28.970","Text":"that the derivative of the sine of something is"},{"Start":"00:28.970 ","End":"00:31.925","Text":"a cosine of something times the internal derivative."},{"Start":"00:31.925 ","End":"00:33.410","Text":"So if we do that at the top,"},{"Start":"00:33.410 ","End":"00:35.090","Text":"we get the cosine of ax,"},{"Start":"00:35.090 ","End":"00:36.965","Text":"but the internal derivative is a."},{"Start":"00:36.965 ","End":"00:40.925","Text":"Likewise for bx, the internal derivative is b, and"},{"Start":"00:40.925 ","End":"00:45.035","Text":"at this point, we can substitute x equals 0 because cosine of 0 is 1."},{"Start":"00:45.035 ","End":"00:48.620","Text":"These are 1 and 1 and all that we\u0027re left with is a over b,"},{"Start":"00:48.620 ","End":"00:49.850","Text":"and that\u0027s this one."},{"Start":"00:49.850 ","End":"00:52.610","Text":"The next one, another case of 0 over 0."},{"Start":"00:52.610 ","End":"00:55.205","Text":"Next is 0, cosine of 0 is 0."},{"Start":"00:55.205 ","End":"00:59.120","Text":"So what we will do is use L\u0027Hôpital, of course,"},{"Start":"00:59.120 ","End":"01:02.315","Text":"and L\u0027Hôpital, we\u0027ll say that instead of doing this limit,"},{"Start":"01:02.315 ","End":"01:06.575","Text":"we\u0027ll do a different limit which is the one where we take the derivatives top and bottom,"},{"Start":"01:06.575 ","End":"01:09.575","Text":"x gives us 1, sine x gives us cosine x,"},{"Start":"01:09.575 ","End":"01:11.390","Text":"on the bottom 3x squared."},{"Start":"01:11.390 ","End":"01:18.255","Text":"Now, we still can\u0027t substitute 0 because we still get 0 over 0 because cosine of 0 is 1."},{"Start":"01:18.255 ","End":"01:21.845","Text":"So we\u0027ll do L\u0027Hôpital again and then if we differentiate the top,"},{"Start":"01:21.845 ","End":"01:26.450","Text":"the 1 goes, minus cosine x becomes plus sine x and we get 6x."},{"Start":"01:26.450 ","End":"01:29.330","Text":"Still no good for substitution, 0 over 0."},{"Start":"01:29.330 ","End":"01:31.640","Text":"Yet again, we\u0027ll go with L\u0027Hôpital."},{"Start":"01:31.640 ","End":"01:35.465","Text":"This time sine x gives us cosine x, 6x gives us 6."},{"Start":"01:35.465 ","End":"01:38.165","Text":"Finally, we can substitute x equals 0."},{"Start":"01:38.165 ","End":"01:40.355","Text":"Remember that cosine of 0 is 1,"},{"Start":"01:40.355 ","End":"01:42.520","Text":"so we end up with just 1/6."},{"Start":"01:42.520 ","End":"01:44.100","Text":"The next one, again,"},{"Start":"01:44.100 ","End":"01:45.765","Text":"when x is 0, this is 0,"},{"Start":"01:45.765 ","End":"01:48.680","Text":"tangent of 0 is 0 and so is sine 0,"},{"Start":"01:48.680 ","End":"01:51.140","Text":"so everything ends up being 0 over 0."},{"Start":"01:51.140 ","End":"01:52.895","Text":"We\u0027ve used L\u0027Hôpital\u0027s rule,"},{"Start":"01:52.895 ","End":"01:56.630","Text":"we get that this is the original exercise and what we do"},{"Start":"01:56.630 ","End":"02:00.440","Text":"is we differentiate the top and we differentiate the bottom."},{"Start":"02:00.440 ","End":"02:04.415","Text":"Remember tangent x is 1 over cosine squared x,"},{"Start":"02:04.415 ","End":"02:06.250","Text":"or cosine to the minus 2."},{"Start":"02:06.250 ","End":"02:09.185","Text":"Sine is cosine, x cubed is 3x squared,"},{"Start":"02:09.185 ","End":"02:13.130","Text":"we\u0027re still with 0 over 0 because cosine of 0 is 1."},{"Start":"02:13.130 ","End":"02:15.650","Text":"So we get 1 minus 1 over 0."},{"Start":"02:15.650 ","End":"02:18.190","Text":"So we need to use L\u0027Hôpital again."},{"Start":"02:18.190 ","End":"02:20.110","Text":"The bottom this time is 6x."},{"Start":"02:20.110 ","End":"02:23.555","Text":"Now, this thing comes out of something to the minus 2."},{"Start":"02:23.555 ","End":"02:27.080","Text":"So we put minus 2 times that something to the minus 3,"},{"Start":"02:27.080 ","End":"02:28.850","Text":"but this is the internal function,"},{"Start":"02:28.850 ","End":"02:31.490","Text":"so we need the internal derivative of cosine,"},{"Start":"02:31.490 ","End":"02:32.870","Text":"which is minus sine x."},{"Start":"02:32.870 ","End":"02:34.940","Text":"Derivative of cosine is minus sine."},{"Start":"02:34.940 ","End":"02:37.985","Text":"So that\u0027s why the minus minus and here, 6x."},{"Start":"02:37.985 ","End":"02:40.100","Text":"We have this first thing is,"},{"Start":"02:40.100 ","End":"02:44.540","Text":"minus with the minus here when multiplied gives us a plus 2."},{"Start":"02:44.540 ","End":"02:50.420","Text":"Now, this is cosine to the minus 3x sine x and here the minus minus becomes a plus,"},{"Start":"02:50.420 ","End":"02:52.820","Text":"if we substitute 0 again,"},{"Start":"02:52.820 ","End":"02:56.660","Text":"since sine of x of 0 is 0 and this is also 0,"},{"Start":"02:56.660 ","End":"02:59.765","Text":"we still have to use L\u0027Hôpital another time."},{"Start":"02:59.765 ","End":"03:02.030","Text":"We see that now the denominator is not 0."},{"Start":"03:02.030 ","End":"03:05.490","Text":"So using the quotient rule and using various rules,"},{"Start":"03:05.490 ","End":"03:08.570","Text":"I\u0027m not going to go into differentiation too deeply now,"},{"Start":"03:08.570 ","End":"03:12.155","Text":"these are side exercises that you can try this at home safely."},{"Start":"03:12.155 ","End":"03:18.785","Text":"So we just substitute x equals 0 at this point and remembering that cosine of 0 is 1,"},{"Start":"03:18.785 ","End":"03:20.165","Text":"every time we see a sine,"},{"Start":"03:20.165 ","End":"03:23.089","Text":"that\u0027s just 0 and here we have only cosines,"},{"Start":"03:23.089 ","End":"03:25.610","Text":"but each of these is 1, 20 power it\u0027s going to be 1."},{"Start":"03:25.610 ","End":"03:27.965","Text":"But this is just 2 plus 1,"},{"Start":"03:27.965 ","End":"03:29.495","Text":"and 2 plus 1 is,"},{"Start":"03:29.495 ","End":"03:30.710","Text":"plus the first bit which is 0,"},{"Start":"03:30.710 ","End":"03:32.960","Text":"like I mentioned and 0 plus 2 plus 1 over 6."},{"Start":"03:32.960 ","End":"03:37.320","Text":"So it\u0027s 3 over 6 and 3 over 6 is equal to 1/2."}],"ID":8453},{"Watched":false,"Name":"Exercises 17(alt way)-18","Duration":"3m 14s","ChapterTopicVideoID":8283,"CourseChapterTopicPlaylistID":52079,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8283.jpeg","UploadDate":"2017-03-02T01:08:01.7470000","DurationForVideoObject":"PT3M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.449","Text":"This exercise is exactly the same as the previous exercise,"},{"Start":"00:03.449 ","End":"00:05.295","Text":"I\u0027m just proposing another solution."},{"Start":"00:05.295 ","End":"00:08.355","Text":"The previous 1 said just using L\u0027Hopital\u0027s rule."},{"Start":"00:08.355 ","End":"00:09.705","Text":"But if we use 2 rules,"},{"Start":"00:09.705 ","End":"00:12.480","Text":"L\u0027Hopital\u0027s rule and the fact that we know that"},{"Start":"00:12.480 ","End":"00:16.500","Text":"this famous limit of sine x over x is 1 when x goes to 0,"},{"Start":"00:16.500 ","End":"00:18.810","Text":"then we can get a bit of a shorter solution."},{"Start":"00:18.810 ","End":"00:20.445","Text":"We\u0027re going to use 2 different rules."},{"Start":"00:20.445 ","End":"00:23.610","Text":"What we\u0027re going to say here is that if we remember"},{"Start":"00:23.610 ","End":"00:27.555","Text":"that tangent of x is sine x over cosine x,"},{"Start":"00:27.555 ","End":"00:31.080","Text":"what we can then do is take sine x out of the top and"},{"Start":"00:31.080 ","End":"00:35.160","Text":"x out of the bottom and what we\u0027ll get is factorizing it this way."},{"Start":"00:35.160 ","End":"00:39.570","Text":"Well, in 2 steps, first of all take sine x out and then sine x over x,"},{"Start":"00:39.570 ","End":"00:41.120","Text":"put x over sine x,"},{"Start":"00:41.120 ","End":"00:43.210","Text":"and put x squared over the rest of it."},{"Start":"00:43.210 ","End":"00:45.690","Text":"The limit of a product is the product of the limit,"},{"Start":"00:45.690 ","End":"00:48.210","Text":"so we can break it off into 2 bits."},{"Start":"00:48.210 ","End":"00:49.650","Text":"At the first 1, as we said,"},{"Start":"00:49.650 ","End":"00:50.995","Text":"is just equal to 1."},{"Start":"00:50.995 ","End":"00:54.320","Text":"So all that we\u0027re left with is this second limit."},{"Start":"00:54.320 ","End":"00:57.080","Text":"Now this is also 0 over 0,"},{"Start":"00:57.080 ","End":"01:01.130","Text":"0 squared is 0, 1 over 1 minus 1 is also 0."},{"Start":"01:01.130 ","End":"01:03.300","Text":"Time again for L\u0027Hopital,"},{"Start":"01:03.300 ","End":"01:08.330","Text":"the 0 over 0 and we get the limit of the derivative of the bottom is 2x,"},{"Start":"01:08.330 ","End":"01:11.450","Text":"a derivative of the top minus 1 doesn\u0027t count for anything."},{"Start":"01:11.450 ","End":"01:14.090","Text":"All we\u0027re left with is the 1 over cosine x,"},{"Start":"01:14.090 ","End":"01:18.695","Text":"and 1 over cosine x should be minus 1 over cosine squared x."},{"Start":"01:18.695 ","End":"01:22.205","Text":"That\u0027s the limit 1 over box minus 1 over box squared."},{"Start":"01:22.205 ","End":"01:26.405","Text":"But the internal derivative of cosine x is minus sine x."},{"Start":"01:26.405 ","End":"01:31.195","Text":"So that minus 1 multiplied by the minus sine x gives us sine x."},{"Start":"01:31.195 ","End":"01:37.405","Text":"Again still 0 over 0 because 2x is 0 and here we have a 0 over 1."},{"Start":"01:37.405 ","End":"01:40.970","Text":"So we can again just throw out the sine x over x,"},{"Start":"01:40.970 ","End":"01:43.550","Text":"because while the 2x splits up into x here,"},{"Start":"01:43.550 ","End":"01:45.710","Text":"and we take the sine x over x,"},{"Start":"01:45.710 ","End":"01:47.795","Text":"that leaves us with 2 in the denominator."},{"Start":"01:47.795 ","End":"01:50.000","Text":"There\u0027s also a 1 over cosine squared x,"},{"Start":"01:50.000 ","End":"01:51.935","Text":"which just adds to the denominator."},{"Start":"01:51.935 ","End":"01:55.475","Text":"This limit here is 1 because it\u0027s sine x over x."},{"Start":"01:55.475 ","End":"01:59.300","Text":"The remaining bit there\u0027s no reason not to substitute x equals 0."},{"Start":"01:59.300 ","End":"02:01.625","Text":"Cosine 0 is 1, 1 over this,"},{"Start":"02:01.625 ","End":"02:03.320","Text":"so it\u0027s 1 times a half,"},{"Start":"02:03.320 ","End":"02:04.445","Text":"which is a half."},{"Start":"02:04.445 ","End":"02:08.435","Text":"Here we have an exercise where L\u0027Hopital will be coming very useful."},{"Start":"02:08.435 ","End":"02:09.590","Text":"It is 0 over 0,"},{"Start":"02:09.590 ","End":"02:10.685","Text":"you can check it."},{"Start":"02:10.685 ","End":"02:14.015","Text":"This is 0, square root of 1 plus 0 is 1,"},{"Start":"02:14.015 ","End":"02:17.210","Text":"minus square root of 1 is also 1, so it is 0 over 0."},{"Start":"02:17.210 ","End":"02:18.845","Text":"If we didn\u0027t have L\u0027Hopital,"},{"Start":"02:18.845 ","End":"02:21.710","Text":"I would bet that you would be using the conjugates."},{"Start":"02:21.710 ","End":"02:24.800","Text":"You would be multiplying top and bottom by this thing with a plus."},{"Start":"02:24.800 ","End":"02:27.470","Text":"But fortunately we do have L\u0027Hopital."},{"Start":"02:27.470 ","End":"02:30.620","Text":"L\u0027Hopital is better than doing it with conjugates."},{"Start":"02:30.620 ","End":"02:34.565","Text":"Now, L\u0027Hopital\u0027s says, instead of this limit compute a different 1,"},{"Start":"02:34.565 ","End":"02:37.700","Text":"which is obtained by differentiating top and bottom."},{"Start":"02:37.700 ","End":"02:39.470","Text":"The bottom is going to be 1 but the top,"},{"Start":"02:39.470 ","End":"02:42.665","Text":"we just need to remember that there is a rule for the square root"},{"Start":"02:42.665 ","End":"02:46.040","Text":"and that the derivative of square root is 1 over twice square root,"},{"Start":"02:46.040 ","End":"02:48.490","Text":"and then there\u0027s the matter of the internal derivative."},{"Start":"02:48.490 ","End":"02:51.170","Text":"What we get, the first 1 is 1 over twice,"},{"Start":"02:51.170 ","End":"02:53.480","Text":"whatever it is times its derivative,"},{"Start":"02:53.480 ","End":"02:55.670","Text":"derivative of 1 plus sine x is cosine x and"},{"Start":"02:55.670 ","End":"02:57.950","Text":"the same with the other,1 over twice whatever it is,"},{"Start":"02:57.950 ","End":"03:00.170","Text":"times its internal derivative."},{"Start":"03:00.170 ","End":"03:03.620","Text":"At this point there\u0027s no reason not to substitute."},{"Start":"03:03.620 ","End":"03:08.000","Text":"So if we put sine of 0 is 0 and cosine of 0 is 1,"},{"Start":"03:08.000 ","End":"03:09.890","Text":"here we get 1 over 2,"},{"Start":"03:09.890 ","End":"03:11.479","Text":"and here we just get nothing,"},{"Start":"03:11.479 ","End":"03:12.770","Text":"and the bottom is 1,"},{"Start":"03:12.770 ","End":"03:15.660","Text":"so the answer is 1/2."}],"ID":8454},{"Watched":false,"Name":"Exercises 19","Duration":"3m 41s","ChapterTopicVideoID":8284,"CourseChapterTopicPlaylistID":52079,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8284.jpeg","UploadDate":"2017-01-25T01:38:41.7670000","DurationForVideoObject":"PT3M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.240","Text":"In this exercise, we have to find this limit,"},{"Start":"00:03.240 ","End":"00:05.490","Text":"this expression as x goes to 0."},{"Start":"00:05.490 ","End":"00:08.160","Text":"If we check by substitution,"},{"Start":"00:08.160 ","End":"00:09.810","Text":"we\u0027ll get 0 over 0."},{"Start":"00:09.810 ","End":"00:16.380","Text":"Denominator is obvious, numerator because 1 minus cosine 0 is 0, this is 0."},{"Start":"00:16.380 ","End":"00:19.800","Text":"Then once again we have 1 minus cosine 0, so it\u0027s still 0,"},{"Start":"00:19.800 ","End":"00:21.110","Text":"so it\u0027s 0 over 0,"},{"Start":"00:21.110 ","End":"00:25.170","Text":"and that\u0027s very often where L\u0027Hopital\u0027s rule helps a lot."},{"Start":"00:25.170 ","End":"00:30.485","Text":"What we get is this but if you get a 0 over 0 case,"},{"Start":"00:30.485 ","End":"00:32.660","Text":"then you can replace this limit with"},{"Start":"00:32.660 ","End":"00:35.270","Text":"a different limit and it will give you the same answer and"},{"Start":"00:35.270 ","End":"00:37.880","Text":"that new limit is what you get when you differentiate"},{"Start":"00:37.880 ","End":"00:41.970","Text":"separately the numerator and then the denominator."},{"Start":"00:41.970 ","End":"00:45.050","Text":"I\u0027m not going to do the whole exercise here,"},{"Start":"00:45.050 ","End":"00:49.640","Text":"but what we basically get is that because the derivative of cosine is minus sine,"},{"Start":"00:49.640 ","End":"00:51.140","Text":"so we get plus here,"},{"Start":"00:51.140 ","End":"00:54.950","Text":"but the 1 disappears because when you differentiate it and so we get"},{"Start":"00:54.950 ","End":"00:59.240","Text":"sine of 1 minus cosine and this will be the internal derivative again,"},{"Start":"00:59.240 ","End":"01:04.655","Text":"the 1 goes and the minus cosine derives to sine and here of course polynomial."},{"Start":"01:04.655 ","End":"01:08.780","Text":"That\u0027s just the derivation of numerator and denominator."},{"Start":"01:08.780 ","End":"01:11.000","Text":"Once again, if we substitute,"},{"Start":"01:11.000 ","End":"01:16.130","Text":"we\u0027ll see that we have a 0 over 0 and the natural tendency follow L\u0027Hopital."},{"Start":"01:16.130 ","End":"01:18.320","Text":"But this is rather messy and there is"},{"Start":"01:18.320 ","End":"01:21.030","Text":"a little bit of a shortcut which often you might see."},{"Start":"01:21.030 ","End":"01:25.370","Text":"Because we know that sine x over x has a limit of 1,"},{"Start":"01:25.370 ","End":"01:29.900","Text":"what we can do is we can take the sine x over x separately,"},{"Start":"01:29.900 ","End":"01:32.360","Text":"and the limit of a product is the product of"},{"Start":"01:32.360 ","End":"01:35.480","Text":"the limit so I can take the sine x over x separately."},{"Start":"01:35.480 ","End":"01:38.690","Text":"Then what we\u0027re left with is this piece here, over here."},{"Start":"01:38.690 ","End":"01:40.415","Text":"After we take the x out,"},{"Start":"01:40.415 ","End":"01:42.305","Text":"then we get 4x squared."},{"Start":"01:42.305 ","End":"01:43.970","Text":"Now, sure we\u0027ll still have to do"},{"Start":"01:43.970 ","End":"01:47.210","Text":"L\u0027Hopital\u0027s rule because if you check here it\u0027s 0 over 0,"},{"Start":"01:47.210 ","End":"01:50.719","Text":"but this expression is much simpler than this expression."},{"Start":"01:50.719 ","End":"01:53.060","Text":"Exactly the numerator is easier."},{"Start":"01:53.060 ","End":"01:58.865","Text":"This we can almost straight away say is 1 and just cancel the first bit,"},{"Start":"01:58.865 ","End":"02:01.310","Text":"which will do but we\u0027re left with the second bit."},{"Start":"02:01.310 ","End":"02:02.390","Text":"Now along the line,"},{"Start":"02:02.390 ","End":"02:04.835","Text":"we\u0027re going to have the derivatives of sine something."},{"Start":"02:04.835 ","End":"02:06.610","Text":"I took the sine x over x out."},{"Start":"02:06.610 ","End":"02:09.200","Text":"The rule that we\u0027re going to remember is when we have"},{"Start":"02:09.200 ","End":"02:11.855","Text":"the sine of something and we differentiate it,"},{"Start":"02:11.855 ","End":"02:13.865","Text":"we get the cosine of the same thing,"},{"Start":"02:13.865 ","End":"02:17.720","Text":"except that we also have to multiply by the internal derivative,"},{"Start":"02:17.720 ","End":"02:19.880","Text":"the derivative of what was in this box."},{"Start":"02:19.880 ","End":"02:22.370","Text":"In this case, just looking back,"},{"Start":"02:22.370 ","End":"02:25.475","Text":"what we get is set at the sine of this thing,"},{"Start":"02:25.475 ","End":"02:28.280","Text":"we get the cosine of the same thing,"},{"Start":"02:28.280 ","End":"02:31.070","Text":"but we have to multiply by the inner derivative of the"},{"Start":"02:31.070 ","End":"02:37.400","Text":"1 minus cosine x and 1 minus cosine x gives us sine x because cosine is minus sine,"},{"Start":"02:37.400 ","End":"02:39.710","Text":"so minus cosine is plus sign."},{"Start":"02:39.710 ","End":"02:44.180","Text":"On the denominator 4 x squared gives us 8x quite immediately."},{"Start":"02:44.180 ","End":"02:48.650","Text":"Actually, we\u0027re once again in the situation where if we put x equals 0,"},{"Start":"02:48.650 ","End":"02:50.765","Text":"we\u0027ll quickly see it\u0027s a 0 over 0."},{"Start":"02:50.765 ","End":"02:53.690","Text":"Once again, we avoid the temptation to immediately"},{"Start":"02:53.690 ","End":"02:57.200","Text":"apply L\u0027Hopital because we have a sine x over x here."},{"Start":"02:57.200 ","End":"03:01.205","Text":"If we separate the sine x over x bit,"},{"Start":"03:01.205 ","End":"03:04.790","Text":"and what we\u0027re left with is this sine x comes out, this x comes out,"},{"Start":"03:04.790 ","End":"03:06.065","Text":"we\u0027re left with the 8 here,"},{"Start":"03:06.065 ","End":"03:09.005","Text":"and we\u0027re left with the cosine of 1 minus cosine x."},{"Start":"03:09.005 ","End":"03:12.605","Text":"Now, the limit of the product as the product of the limits so that\u0027s okay."},{"Start":"03:12.605 ","End":"03:18.035","Text":"But this limit of sine x over x is the well-known famous limit that comes out to 1,"},{"Start":"03:18.035 ","End":"03:19.730","Text":"so this we can throw off,"},{"Start":"03:19.730 ","End":"03:21.890","Text":"and what we get is just this bit,"},{"Start":"03:21.890 ","End":"03:23.480","Text":"and this already is no longer 8,"},{"Start":"03:23.480 ","End":"03:24.770","Text":"so anything go over 0."},{"Start":"03:24.770 ","End":"03:29.360","Text":"We can substitute x equals 0 and x goes to 0."},{"Start":"03:29.360 ","End":"03:35.480","Text":"We already said that 1 minus cosine of 0 is 0 and cosine 0 is 1."},{"Start":"03:35.480 ","End":"03:37.550","Text":"What we\u0027re left with is the numerator is 1,"},{"Start":"03:37.550 ","End":"03:42.360","Text":"denominator is 8, so the answer to this 1 is just 1/8."}],"ID":8455},{"Watched":false,"Name":"Exercises 20","Duration":"3m 13s","ChapterTopicVideoID":8285,"CourseChapterTopicPlaylistID":52079,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8285.jpeg","UploadDate":"2017-01-25T01:39:36.1700000","DurationForVideoObject":"PT3M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.565","Text":"The last exercise in this series of L\u0027Hopital 0 over 0 limits."},{"Start":"00:05.565 ","End":"00:07.320","Text":"Here it is repeated."},{"Start":"00:07.320 ","End":"00:09.525","Text":"What we have is this limit."},{"Start":"00:09.525 ","End":"00:12.255","Text":"If we substitute x equals 0,"},{"Start":"00:12.255 ","End":"00:18.510","Text":"we see that the denominator is 0 and so is the numerator because the sine of 0 is 0."},{"Start":"00:18.510 ","End":"00:20.970","Text":"That takes care of this part and x is 0,"},{"Start":"00:20.970 ","End":"00:22.290","Text":"takes care of the other part."},{"Start":"00:22.290 ","End":"00:26.070","Text":"We have a 0 over 0 situation and the obvious thing"},{"Start":"00:26.070 ","End":"00:29.820","Text":"to do is L\u0027Hopital and derivative of this is this."},{"Start":"00:29.820 ","End":"00:34.065","Text":"Here we have a product and if we use the product rule gives us these 2 terms."},{"Start":"00:34.065 ","End":"00:39.680","Text":"If we just expand the polynomial minus x minus x squared and derive it, we get this part."},{"Start":"00:39.680 ","End":"00:41.930","Text":"Up to now, we\u0027ve just decide that this,"},{"Start":"00:41.930 ","End":"00:45.200","Text":"from 0 over 0 equals by L\u0027Hopital to this."},{"Start":"00:45.200 ","End":"00:46.880","Text":"Remember what L\u0027Hopital\u0027s said,"},{"Start":"00:46.880 ","End":"00:48.815","Text":"it\u0027s not that this is equal to this,"},{"Start":"00:48.815 ","End":"00:51.140","Text":"but that if we differentiate the numerator and"},{"Start":"00:51.140 ","End":"00:54.455","Text":"the denominator separately on a 0 over 0 case,"},{"Start":"00:54.455 ","End":"00:55.685","Text":"we get a different limit,"},{"Start":"00:55.685 ","End":"00:58.895","Text":"but the answer will be the same as the original limit."},{"Start":"00:58.895 ","End":"01:01.190","Text":"Now we have this to deal with."},{"Start":"01:01.190 ","End":"01:04.084","Text":"If we substitute x equals 0,"},{"Start":"01:04.084 ","End":"01:07.595","Text":"and again, you\u0027ll see that we get 0 over 0."},{"Start":"01:07.595 ","End":"01:08.840","Text":"If you compute it,"},{"Start":"01:08.840 ","End":"01:10.820","Text":"this is 0 because of the sign,"},{"Start":"01:10.820 ","End":"01:12.455","Text":"this is 0 obviously,"},{"Start":"01:12.455 ","End":"01:15.510","Text":"and the cosine of 0 is 1."},{"Start":"01:16.030 ","End":"01:19.040","Text":"All of this is 0, cosine of it is 1,"},{"Start":"01:19.040 ","End":"01:22.520","Text":"less 1 is 0, and also this is 0 because x is 0."},{"Start":"01:22.520 ","End":"01:28.220","Text":"We are again in a 0 over 0 situation and we\u0027ll just use L\u0027Hopital\u0027s."},{"Start":"01:28.220 ","End":"01:31.580","Text":"Differentiating the bottom gives us 6x."},{"Start":"01:31.580 ","End":"01:33.575","Text":"If we differentiate the top,"},{"Start":"01:33.575 ","End":"01:35.870","Text":"then from these 2 terms,"},{"Start":"01:35.870 ","End":"01:39.305","Text":"we just get the minus 2."},{"Start":"01:39.305 ","End":"01:43.220","Text":"Each of these gives us from the product rule 2 terms."},{"Start":"01:43.220 ","End":"01:45.920","Text":"This 1 gives us like the f prime g,"},{"Start":"01:45.920 ","End":"01:48.610","Text":"and this gives us the fg prime."},{"Start":"01:48.610 ","End":"01:54.230","Text":"We can simplify this a bit because this term and this term both"},{"Start":"01:54.230 ","End":"02:00.020","Text":"give us twice e^x cosine x or cosine xe^x."},{"Start":"02:00.020 ","End":"02:02.360","Text":"This term is the same as this term,"},{"Start":"02:02.360 ","End":"02:04.430","Text":"the product\u0027s just written in a backward order,"},{"Start":"02:04.430 ","End":"02:07.190","Text":"but it\u0027s cosine x times e^x or the reverse."},{"Start":"02:07.190 ","End":"02:08.240","Text":"The same thing here,"},{"Start":"02:08.240 ","End":"02:12.020","Text":"we have an e^x and we have a sine x only this time there\u0027s a minus here,"},{"Start":"02:12.020 ","End":"02:14.255","Text":"so these 2 cancel each other out,"},{"Start":"02:14.255 ","End":"02:15.485","Text":"and these 2 together."},{"Start":"02:15.485 ","End":"02:18.350","Text":"You can combine them into 1 with a 2 in front of it."},{"Start":"02:18.350 ","End":"02:21.575","Text":"That\u0027s practically what\u0027s on the next line."},{"Start":"02:21.575 ","End":"02:23.005","Text":"Just simplified."},{"Start":"02:23.005 ","End":"02:25.880","Text":"Let\u0027s take 2 out of the brackets."},{"Start":"02:25.880 ","End":"02:27.874","Text":"That might be a bit simpler."},{"Start":"02:27.874 ","End":"02:31.325","Text":"We\u0027ve got this minus 1 over the 3x."},{"Start":"02:31.325 ","End":"02:34.220","Text":"What we can do with this is again, it\u0027s L\u0027Hopital,"},{"Start":"02:34.220 ","End":"02:38.630","Text":"this is 0, this is cosine 0 minus 1, so 1 minus 1."},{"Start":"02:38.630 ","End":"02:43.850","Text":"Once again, L\u0027Hopital, the 3x becomes 3 when we do the denominator."},{"Start":"02:43.850 ","End":"02:45.115","Text":"Then we\u0027ll do the numerator,"},{"Start":"02:45.115 ","End":"02:49.610","Text":"the minus 1 disappears and we just have a product cosine of x times e^x."},{"Start":"02:49.610 ","End":"02:52.565","Text":"With the product rule, we get 1 of the things derived,"},{"Start":"02:52.565 ","End":"02:54.740","Text":"the other not derived, and vice versa."},{"Start":"02:54.740 ","End":"02:56.090","Text":"We\u0027re left with this."},{"Start":"02:56.090 ","End":"02:59.240","Text":"Now, at this point, we can substitute just x equals 0."},{"Start":"02:59.240 ","End":"03:03.245","Text":"There is no problem. The 3 stays 3 and the numerator,"},{"Start":"03:03.245 ","End":"03:06.290","Text":"when x is 0, sine x is 0, that disappears."},{"Start":"03:06.290 ","End":"03:08.405","Text":"When x is 0, this is 1,"},{"Start":"03:08.405 ","End":"03:09.950","Text":"and this is 1, so it\u0027s 1."},{"Start":"03:09.950 ","End":"03:14.070","Text":"We\u0027re left with 1 over 3 and that\u0027s the answer."}],"ID":8456},{"Watched":false,"Name":"Exercise 21","Duration":"7m 6s","ChapterTopicVideoID":1336,"CourseChapterTopicPlaylistID":52079,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1336.jpeg","UploadDate":"2014-10-01T04:49:03.5830000","DurationForVideoObject":"PT7M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:10.485","Text":"[inaudible] As x goes to 0 of this expression,"},{"Start":"00:10.485 ","End":"00:15.210","Text":"the first thing to do is see what goes wrong with substitution."},{"Start":"00:15.210 ","End":"00:17.100","Text":"Put x equals 0 here,"},{"Start":"00:17.100 ","End":"00:19.590","Text":"we get 0 on the denominator."},{"Start":"00:19.590 ","End":"00:23.790","Text":"0 squared is 0, cosine 0 is 1,"},{"Start":"00:23.790 ","End":"00:25.995","Text":"log of 1 is 0,"},{"Start":"00:25.995 ","End":"00:31.350","Text":"so it\u0027s 0 over 0."},{"Start":"00:31.350 ","End":"00:40.090","Text":"The solution just repeated the original exercise. Scroll up a bit."},{"Start":"00:40.160 ","End":"00:47.370","Text":"The natural thing to do would be L\u0027Hopital and just again,"},{"Start":"00:47.370 ","End":"00:56.700","Text":"a reminder of what L\u0027Hopital said and occasionally write his name in his honor."},{"Start":"00:56.700 ","End":"01:01.785","Text":"He was a clever French mathematician, so L\u0027Hopital."},{"Start":"01:01.785 ","End":"01:09.855","Text":"Basically, what he said is mostly related to cases of 0 over 0 or infinity over infinity."},{"Start":"01:09.855 ","End":"01:14.340","Text":"He said that in cases of limits which when you substitute look like 1 of this."},{"Start":"01:14.340 ","End":"01:18.170","Text":"What you do is instead of computing the original limit,"},{"Start":"01:18.170 ","End":"01:20.240","Text":"you compute a different limit,"},{"Start":"01:20.240 ","End":"01:26.705","Text":"the 1 which is obtained by separately differentiating the numerator and the denominator."},{"Start":"01:26.705 ","End":"01:29.000","Text":"It\u0027s very helpful in many,"},{"Start":"01:29.000 ","End":"01:34.100","Text":"many cases of 0 over 0 and it\u0027s a real shortcut,"},{"Start":"01:34.100 ","End":"01:40.660","Text":"and yeah, very worthwhile contribution."},{"Start":"01:40.970 ","End":"01:43.950","Text":"What is this other limit?"},{"Start":"01:43.950 ","End":"01:47.715","Text":"It\u0027s when we differentiate this and this separately,"},{"Start":"01:47.715 ","End":"01:51.345","Text":"but we\u0027re going to need a formula to remember what is"},{"Start":"01:51.345 ","End":"01:57.345","Text":"the derivative of the natural log of something."},{"Start":"01:57.345 ","End":"02:00.900","Text":"Normally if it\u0027s natural log of x is 1 over x."},{"Start":"02:00.900 ","End":"02:02.460","Text":"But if that x is not just an x,"},{"Start":"02:02.460 ","End":"02:03.810","Text":"but a whole something,"},{"Start":"02:03.810 ","End":"02:06.780","Text":"then it\u0027s 1 over something times an extra bit,"},{"Start":"02:06.780 ","End":"02:09.150","Text":"which is called the internal derivative or"},{"Start":"02:09.150 ","End":"02:12.630","Text":"the derivative of the internal function of whatever that was."},{"Start":"02:12.630 ","End":"02:15.585","Text":"In our case, it\u0027s going to be cosine of x squared."},{"Start":"02:15.585 ","End":"02:19.845","Text":"So that\u0027s if we use this rule,"},{"Start":"02:19.845 ","End":"02:25.770","Text":"then what we should get is that it\u0027s isn\u0027t just 1 over cosine squared x,"},{"Start":"02:25.770 ","End":"02:28.080","Text":"it\u0027s times the internal derivative which"},{"Start":"02:28.080 ","End":"02:32.175","Text":"we haven\u0027t computed yet just indicated it, still to be done."},{"Start":"02:32.175 ","End":"02:42.255","Text":"The denominator is x to the 4."},{"Start":"02:42.255 ","End":"02:48.810","Text":"I think we should say here derivative because if we\u0027re deriving the denominator,"},{"Start":"02:48.810 ","End":"02:53.415","Text":"then what we need to add here is this thing."},{"Start":"02:53.415 ","End":"02:56.460","Text":"I\u0027m just having a little trouble, technical trouble here,"},{"Start":"02:56.460 ","End":"02:58.380","Text":"just a second."},{"Start":"02:58.380 ","End":"03:01.810","Text":"Yes, derivative."},{"Start":"03:02.540 ","End":"03:11.050","Text":"Then we go ahead and do those derivatives."},{"Start":"03:16.640 ","End":"03:21.750","Text":"Remember that when we need to differentiate cosine of x squared,"},{"Start":"03:21.750 ","End":"03:26.730","Text":"this is also a chain rule but with cosine being the outer function."},{"Start":"03:26.730 ","End":"03:33.090","Text":"Since the derivative of cosine of plain x is minus sine x, if instead of x,"},{"Start":"03:33.090 ","End":"03:35.775","Text":"it\u0027s a complicated thing, an internal function,"},{"Start":"03:35.775 ","End":"03:40.870","Text":"then we need to say minus sine of that something but times its derivative."},{"Start":"03:54.560 ","End":"04:00.645","Text":"What we\u0027re going to do is do these 2 derivatives to place where it says, prime."},{"Start":"04:00.645 ","End":"04:09.540","Text":"So the cosine of something derivative in this case it\u0027s cosine x squared derivative,"},{"Start":"04:09.540 ","End":"04:10.620","Text":"is first of all,"},{"Start":"04:10.620 ","End":"04:13.155","Text":"the minus sign which comes from the cosine."},{"Start":"04:13.155 ","End":"04:17.235","Text":"So we start off with minus sine of x squared,"},{"Start":"04:17.235 ","End":"04:24.935","Text":"but this x squared is the internal function and its derivative is 2x."},{"Start":"04:24.935 ","End":"04:31.910","Text":"This is what we get at this point and now we\u0027ve done all the differentiating."},{"Start":"04:31.910 ","End":"04:40.350","Text":"The next thing to do is to simplify a bit."},{"Start":"04:43.570 ","End":"04:49.860","Text":"Now here is quite a bit of simplification or rearrangement will do."},{"Start":"04:51.050 ","End":"04:56.880","Text":"First of all, I\u0027ll show you where I\u0027m heading and I\u0027m heading towards this."},{"Start":"04:56.880 ","End":"05:05.590","Text":"Now, what is happening here is that sine of x squared over x squared."},{"Start":"05:05.590 ","End":"05:10.220","Text":"These 2 pieces together are very good because they look very much"},{"Start":"05:10.220 ","End":"05:16.510","Text":"like sine x over x as x goes to 0 and that we know."},{"Start":"05:16.510 ","End":"05:20.810","Text":"If I take this separately and also"},{"Start":"05:20.810 ","End":"05:24.920","Text":"the sum cancellation x with x cubed becomes just x squared,"},{"Start":"05:24.920 ","End":"05:27.710","Text":"the 2 goes into the 4 twice,"},{"Start":"05:27.710 ","End":"05:31.205","Text":"there\u0027s still a minus and this thing goes downstairs."},{"Start":"05:31.205 ","End":"05:35.840","Text":"What we have is this and if I rewrite it,"},{"Start":"05:35.840 ","End":"05:42.210","Text":"and remember that also the derivative of a product is the product of the derivatives,"},{"Start":"05:42.210 ","End":"05:47.050","Text":"so I\u0027ll take separately the sine x squared over x squared."},{"Start":"05:48.770 ","End":"05:52.680","Text":"There\u0027s a sine x squared over x squared and"},{"Start":"05:52.680 ","End":"05:56.475","Text":"then the rest of it is minus 1 over 2 cosine of x squared."},{"Start":"05:56.475 ","End":"06:03.450","Text":"Now this limit goes to 0 because x squared could be some other quantity t,"},{"Start":"06:03.450 ","End":"06:07.635","Text":"and when x goes to 0 also t goes to 0 and vice versa."},{"Start":"06:07.635 ","End":"06:10.785","Text":"So basically this limit is just 1."},{"Start":"06:10.785 ","End":"06:13.155","Text":"It\u0027s like the sine x over x limit."},{"Start":"06:13.155 ","End":"06:18.990","Text":"This can disappear and all we\u0027re left with is this limit here."},{"Start":"06:18.990 ","End":"06:20.895","Text":"Now this 2nd limit,"},{"Start":"06:20.895 ","End":"06:24.180","Text":"none will trouble, no need for L\u0027Hopital."},{"Start":"06:24.180 ","End":"06:26.820","Text":"We can just do a straight forward substitution."},{"Start":"06:26.820 ","End":"06:33.225","Text":"If I put x equals 0, 0 squared is 0,"},{"Start":"06:33.225 ","End":"06:38.050","Text":"cosine of 0 is 1."},{"Start":"06:38.630 ","End":"06:49.260","Text":"If this is 1 then all I\u0027m left with is minus 1 over 2 times that 1."},{"Start":"06:49.260 ","End":"06:56.090","Text":"Basically, that\u0027s the answer except that this would lead in a bit of"},{"Start":"06:56.090 ","End":"07:03.980","Text":"a tidier form that it\u0027s minus 1/2 and that\u0027s the answer to this 1,"},{"Start":"07:03.980 ","End":"07:07.470","Text":"and this exercise is done."}],"ID":1440},{"Watched":false,"Name":"Exercises 22","Duration":"2m 38s","ChapterTopicVideoID":8286,"CourseChapterTopicPlaylistID":52079,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8286.jpeg","UploadDate":"2017-01-25T01:40:23.2870000","DurationForVideoObject":"PT2M38S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.530","Text":"In this exercise, we have to figure out the limit as x goes to 0"},{"Start":"00:04.530 ","End":"00:09.150","Text":"of this expression with a couple of inverse trigonometrical functions."},{"Start":"00:09.150 ","End":"00:11.400","Text":"You probably guessed that this is 1 of these"},{"Start":"00:11.400 ","End":"00:14.460","Text":"L\u0027Hospital\u0027s 0 over 0 because we\u0027ve been doing those,"},{"Start":"00:14.460 ","End":"00:15.765","Text":"but let\u0027s just check."},{"Start":"00:15.765 ","End":"00:19.050","Text":"We try to substitute x equals 0 on the denominator."},{"Start":"00:19.050 ","End":"00:21.840","Text":"This thing is 0, this part is 0,"},{"Start":"00:21.840 ","End":"00:26.340","Text":"and the arc tangent of 0 is 0 because the tangent of 0 is 0,"},{"Start":"00:26.340 ","End":"00:29.280","Text":"and it\u0027s just the inverse and the same thing with the arcsine."},{"Start":"00:29.280 ","End":"00:31.305","Text":"Arcsine of 0 is also 0."},{"Start":"00:31.305 ","End":"00:33.840","Text":"The sine of that 0 is the first 0,"},{"Start":"00:33.840 ","End":"00:36.135","Text":"so we are indeed in the L\u0027Hospital thing,"},{"Start":"00:36.135 ","End":"00:41.690","Text":"and so we\u0027re going to do what he discovered is that if instead of computing this limit,"},{"Start":"00:41.690 ","End":"00:43.235","Text":"we compute a different limit,"},{"Start":"00:43.235 ","End":"00:44.660","Text":"which is not the original,"},{"Start":"00:44.660 ","End":"00:47.060","Text":"but what happens when you differentiate the top"},{"Start":"00:47.060 ","End":"00:49.550","Text":"and you differentiate the bottom separately?"},{"Start":"00:49.550 ","End":"00:53.105","Text":"That sometimes turns out to be an easier limit. It usually does."},{"Start":"00:53.105 ","End":"00:56.210","Text":"Again, another thing I have to say before we continue is"},{"Start":"00:56.210 ","End":"01:00.180","Text":"the formulae for the derivatives of the arctangent and the arcsine,"},{"Start":"01:00.180 ","End":"01:02.160","Text":"so I\u0027ve written them here in the box,"},{"Start":"01:02.160 ","End":"01:04.205","Text":"so we can use those as needed."},{"Start":"01:04.205 ","End":"01:07.035","Text":"Let\u0027s take the top part first."},{"Start":"01:07.035 ","End":"01:08.950","Text":"Here, we use the arctangent."},{"Start":"01:08.950 ","End":"01:12.695","Text":"What we need is the arctangent of something."},{"Start":"01:12.695 ","End":"01:16.470","Text":"First of all, it\u0027s 1 over 1 plus that something squared,"},{"Start":"01:16.470 ","End":"01:17.960","Text":"and that\u0027s this bit here."},{"Start":"01:17.960 ","End":"01:20.945","Text":"Now, the box is just to remind us that if it\u0027s not x,"},{"Start":"01:20.945 ","End":"01:22.280","Text":"if it\u0027s a function of x,"},{"Start":"01:22.280 ","End":"01:23.735","Text":"then we\u0027re using the chain rule,"},{"Start":"01:23.735 ","End":"01:26.030","Text":"and we\u0027ve just done the outer derivative,"},{"Start":"01:26.030 ","End":"01:28.130","Text":"and so we need to do also the inner."},{"Start":"01:28.130 ","End":"01:31.625","Text":"The inner is what happens when we differentiate what\u0027s in the box."},{"Start":"01:31.625 ","End":"01:34.220","Text":"In this case, it\u0027s x squared plus 3x,"},{"Start":"01:34.220 ","End":"01:36.380","Text":"and we\u0027ll just leave it with a prime as is,"},{"Start":"01:36.380 ","End":"01:38.210","Text":"and we\u0027ll differentiate it in a moment."},{"Start":"01:38.210 ","End":"01:39.890","Text":"We\u0027re just using this formulae."},{"Start":"01:39.890 ","End":"01:41.450","Text":"On the bottom, it\u0027s arcsine,"},{"Start":"01:41.450 ","End":"01:42.770","Text":"so instead of this formula,"},{"Start":"01:42.770 ","End":"01:44.465","Text":"we\u0027re going to be taking this formula,"},{"Start":"01:44.465 ","End":"01:50.030","Text":"and this is the expression we get times the inner derivative which I left with the prime."},{"Start":"01:50.030 ","End":"01:51.650","Text":"What we have to do is, first of all,"},{"Start":"01:51.650 ","End":"01:54.155","Text":"just differentiate where it says prime."},{"Start":"01:54.155 ","End":"01:56.480","Text":"X squared plus 3x, 2x plus 3,"},{"Start":"01:56.480 ","End":"01:57.920","Text":"x squared minus 4x,"},{"Start":"01:57.920 ","End":"02:00.030","Text":"clearly 2x minus 4."},{"Start":"02:00.030 ","End":"02:01.775","Text":"Now, we\u0027ve differentiated it,"},{"Start":"02:01.775 ","End":"02:03.185","Text":"but it looks a bit of a mess."},{"Start":"02:03.185 ","End":"02:05.300","Text":"Let\u0027s see what happens if we put x equals 0,"},{"Start":"02:05.300 ","End":"02:06.785","Text":"maybe we can just substitute."},{"Start":"02:06.785 ","End":"02:10.100","Text":"Well, if x is 0, then this thing is 0,"},{"Start":"02:10.100 ","End":"02:13.010","Text":"all squared is 0, so we get 1 over 1."},{"Start":"02:13.010 ","End":"02:17.705","Text":"So this part is 1 and 2x plus 3 is twice 0 plus 3."},{"Start":"02:17.705 ","End":"02:21.110","Text":"We can just substitute numbers here, these on the top."},{"Start":"02:21.110 ","End":"02:22.550","Text":"We do the same thing on the bottom."},{"Start":"02:22.550 ","End":"02:24.230","Text":"This will come out 0,"},{"Start":"02:24.230 ","End":"02:27.510","Text":"this will come out 1, twice 0 minus 4 is minus 4."},{"Start":"02:27.510 ","End":"02:30.630","Text":"For the top, we get 3 and for the bottom,"},{"Start":"02:30.630 ","End":"02:31.930","Text":"we get minus 4."},{"Start":"02:31.930 ","End":"02:33.125","Text":"So all in all,"},{"Start":"02:33.125 ","End":"02:35.710","Text":"the answer is just 3 over minus 4."},{"Start":"02:35.710 ","End":"02:38.970","Text":"That\u0027s the answer for this exercise."}],"ID":8457},{"Watched":false,"Name":"Exercises 23-24","Duration":"3m 11s","ChapterTopicVideoID":8287,"CourseChapterTopicPlaylistID":52079,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8287.jpeg","UploadDate":"2017-03-02T01:08:05.0130000","DurationForVideoObject":"PT3M11S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.180","Text":"Here we have to find the limit as x goes to 0 of"},{"Start":"00:03.180 ","End":"00:06.945","Text":"the sine of x over the hyperbolic sine of x."},{"Start":"00:06.945 ","End":"00:09.000","Text":"If we substitute x equals 0,"},{"Start":"00:09.000 ","End":"00:10.800","Text":"the sine of x is 0,"},{"Start":"00:10.800 ","End":"00:13.920","Text":"and also the hyperbolic sine of x is 0,"},{"Start":"00:13.920 ","End":"00:16.560","Text":"so it is 0 over 0,"},{"Start":"00:16.560 ","End":"00:19.470","Text":"which brings to mind L\u0027Hopital and his rule,"},{"Start":"00:19.470 ","End":"00:22.410","Text":"and what we\u0027re going to do is find a different limit"},{"Start":"00:22.410 ","End":"00:25.790","Text":"which is obtained by differentiating both top and bottom,"},{"Start":"00:25.790 ","End":"00:28.410","Text":"and it will give us the same answer as the original limit."},{"Start":"00:28.410 ","End":"00:31.130","Text":"What we get is the following."},{"Start":"00:31.130 ","End":"00:33.395","Text":"Now, you should remember some of these formulae."},{"Start":"00:33.395 ","End":"00:36.890","Text":"That the derivative of sine is cosine,"},{"Start":"00:36.890 ","End":"00:41.210","Text":"and the derivative of sine hyperbolic is cosine hyperbolic."},{"Start":"00:41.210 ","End":"00:42.440","Text":"This is the new limit,"},{"Start":"00:42.440 ","End":"00:45.050","Text":"we\u0027re going to compute the L\u0027Hopital 0 over 0."},{"Start":"00:45.050 ","End":"00:48.650","Text":"X goes to 0, so the obvious thing to try is just substituting,"},{"Start":"00:48.650 ","End":"00:51.320","Text":"and cosine of 0 is 1."},{"Start":"00:51.320 ","End":"00:54.745","Text":"But also the cosine hyperbolic of 0 is 1."},{"Start":"00:54.745 ","End":"01:00.290","Text":"Just to briefly remind you this is e to the x plus e to the minus x over 2."},{"Start":"01:00.290 ","End":"01:04.130","Text":"Well, an e to the x and e to the minus x are both 1 and x is 0,"},{"Start":"01:04.130 ","End":"01:05.975","Text":"so we get 1 plus 1 over 2,"},{"Start":"01:05.975 ","End":"01:07.100","Text":"which is also 1."},{"Start":"01:07.100 ","End":"01:08.600","Text":"At the bottom, 1 over 1,"},{"Start":"01:08.600 ","End":"01:11.000","Text":"which equals 1, so that\u0027s this one."},{"Start":"01:11.000 ","End":"01:12.290","Text":"In the next exercise,"},{"Start":"01:12.290 ","End":"01:15.560","Text":"as usual we first check what the problem."},{"Start":"01:15.560 ","End":"01:17.825","Text":"If there is any by substituting,"},{"Start":"01:17.825 ","End":"01:20.014","Text":"if x is is 0, cosine hyperbolic,"},{"Start":"01:20.014 ","End":"01:22.695","Text":"regular cosine of 0 is 1,"},{"Start":"01:22.695 ","End":"01:24.735","Text":"twice 0 is 0, of course also."},{"Start":"01:24.735 ","End":"01:26.310","Text":"1 minus 1 is 0,"},{"Start":"01:26.310 ","End":"01:27.915","Text":"that\u0027s the denominator 0,"},{"Start":"01:27.915 ","End":"01:33.020","Text":"and the top part also cosine hyperbolic of 0 is also 1,"},{"Start":"01:33.020 ","End":"01:35.180","Text":"so we get twice 1 minus 2,"},{"Start":"01:35.180 ","End":"01:36.515","Text":"and that\u0027s also 0."},{"Start":"01:36.515 ","End":"01:38.240","Text":"We have a 0 over 0,"},{"Start":"01:38.240 ","End":"01:41.120","Text":"and so like before we take a different limit which"},{"Start":"01:41.120 ","End":"01:44.149","Text":"is obtained by differentiating top and bottom,"},{"Start":"01:44.149 ","End":"01:48.679","Text":"and what we get if we do this is the minus 2 disappears, it\u0027s a constant,"},{"Start":"01:48.679 ","End":"01:53.925","Text":"the 2 stays, and the derivative of cosine hyperbolic is sine hyperbolic."},{"Start":"01:53.925 ","End":"01:55.820","Text":"You don\u0027t get this alternation or with"},{"Start":"01:55.820 ","End":"01:59.090","Text":"the regular trig functions where sine goes to cosine,"},{"Start":"01:59.090 ","End":"02:00.935","Text":"but cosine goes to minus sine."},{"Start":"02:00.935 ","End":"02:03.320","Text":"Here it just alternate, twice sine hyperbolic."},{"Start":"02:03.320 ","End":"02:06.575","Text":"Like I mentioned, for cosine we get minus sine,"},{"Start":"02:06.575 ","End":"02:08.450","Text":"and the minus together make it a plus,"},{"Start":"02:08.450 ","End":"02:10.085","Text":"so you don\u0027t see any minus here."},{"Start":"02:10.085 ","End":"02:12.470","Text":"The 1 of course drops off because it\u0027s a constant,"},{"Start":"02:12.470 ","End":"02:17.020","Text":"and the derivative of cosine 2x would be sine 2x."},{"Start":"02:17.020 ","End":"02:19.550","Text":"It wasn\u0027t for the chain rule that says we have to also"},{"Start":"02:19.550 ","End":"02:22.670","Text":"multiply by the inner derivative, which is 2."},{"Start":"02:22.670 ","End":"02:24.245","Text":"All we have to do now,"},{"Start":"02:24.245 ","End":"02:25.920","Text":"well, we can cancel the 2s."},{"Start":"02:25.920 ","End":"02:29.720","Text":"Now, we\u0027re also in a 0 over 0 situation because as I said,"},{"Start":"02:29.720 ","End":"02:31.400","Text":"the sine of 0 is 0,"},{"Start":"02:31.400 ","End":"02:33.280","Text":"and so is the sine hyperbolic,"},{"Start":"02:33.280 ","End":"02:35.360","Text":"so we have to use L\u0027Hopital again,"},{"Start":"02:35.360 ","End":"02:38.475","Text":"but at the same time we\u0027ll also throw off the 2s."},{"Start":"02:38.475 ","End":"02:41.120","Text":"Sine becomes cosine hyperbolic,"},{"Start":"02:41.120 ","End":"02:42.320","Text":"and so as I say,"},{"Start":"02:42.320 ","End":"02:45.900","Text":"the sine hyperbolic goes to the cosine hyperbolic,"},{"Start":"02:45.900 ","End":"02:48.840","Text":"and regular sine goes to cosine."},{"Start":"02:48.840 ","End":"02:51.470","Text":"But again, we have an internal function 2x,"},{"Start":"02:51.470 ","End":"02:54.560","Text":"so its derivative is 2 and that\u0027s the chain rule."},{"Start":"02:54.560 ","End":"02:57.230","Text":"At this point, substitution will actually work."},{"Start":"02:57.230 ","End":"02:59.015","Text":"We have a 2 here, now,"},{"Start":"02:59.015 ","End":"03:01.860","Text":"twice 0 is 0 and cosine of 0 is 1,"},{"Start":"03:01.860 ","End":"03:04.070","Text":"so all this denominator is a 2,"},{"Start":"03:04.070 ","End":"03:06.900","Text":"and the cosine hyperbolic of 0 is 1,"},{"Start":"03:06.900 ","End":"03:12.930","Text":"and so the final answer to this exercise is 1/2, and that\u0027s it."}],"ID":8458},{"Watched":false,"Name":"Exercises 25-26","Duration":"3m 3s","ChapterTopicVideoID":8288,"CourseChapterTopicPlaylistID":52079,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8288.jpeg","UploadDate":"2017-03-02T01:08:08.2570000","DurationForVideoObject":"PT3M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.329","Text":"In this exercise, we have to find the limit as x goes to infinity of this expression,"},{"Start":"00:06.329 ","End":"00:10.005","Text":"a polynomial over a polynomial sometimes called a rational function."},{"Start":"00:10.005 ","End":"00:15.075","Text":"As usual, we try to first substitute and see if anything goes wrong."},{"Start":"00:15.075 ","End":"00:18.480","Text":"We can\u0027t substitute, but we can find the limit as"},{"Start":"00:18.480 ","End":"00:21.975","Text":"x goes to infinity of the top and of the bottom."},{"Start":"00:21.975 ","End":"00:24.375","Text":"When we take x goes to infinity,"},{"Start":"00:24.375 ","End":"00:27.525","Text":"then we see that the denominator also goes to infinity."},{"Start":"00:27.525 ","End":"00:31.020","Text":"Actually, all you have to do is look at the highest power term,"},{"Start":"00:31.020 ","End":"00:32.295","Text":"which is 2x squared,"},{"Start":"00:32.295 ","End":"00:34.020","Text":"and when x goes to infinity,"},{"Start":"00:34.020 ","End":"00:35.430","Text":"which is plus infinity,"},{"Start":"00:35.430 ","End":"00:37.850","Text":"then x squared also goes to plus infinity."},{"Start":"00:37.850 ","End":"00:39.380","Text":"As a matter of fact, in this case,"},{"Start":"00:39.380 ","End":"00:41.240","Text":"even if x went to minus infinity,"},{"Start":"00:41.240 ","End":"00:42.830","Text":"this would still go to plus infinity,"},{"Start":"00:42.830 ","End":"00:44.750","Text":"but I just mentioned it because it\u0027s not always,"},{"Start":"00:44.750 ","End":"00:46.970","Text":"sometimes it can be plus or minus."},{"Start":"00:46.970 ","End":"00:48.200","Text":"That\u0027s infinity."},{"Start":"00:48.200 ","End":"00:49.250","Text":"The same thing at the top,"},{"Start":"00:49.250 ","End":"00:51.140","Text":"the highest power is the x squared,"},{"Start":"00:51.140 ","End":"00:53.120","Text":"so when x goes to infinity,"},{"Start":"00:53.120 ","End":"00:55.315","Text":"x squared also goes to infinity."},{"Start":"00:55.315 ","End":"00:58.785","Text":"Here we get infinity over infinity"},{"Start":"00:58.785 ","End":"01:02.300","Text":"this is the first time we\u0027ve seen infinity over infinity in this series,"},{"Start":"01:02.300 ","End":"01:04.145","Text":"we\u0027ve always had 0 over 0."},{"Start":"01:04.145 ","End":"01:08.735","Text":"I\u0027d just like to briefly mention the L\u0027Hopital, which is written."},{"Start":"01:08.735 ","End":"01:12.155","Text":"Basically, he made rules for what to do with limits,"},{"Start":"01:12.155 ","End":"01:17.690","Text":"which come out to be either 0 over 0 or infinity over infinity."},{"Start":"01:17.690 ","End":"01:22.595","Text":"In both cases, what he said was that if we don\u0027t compute the original limit,"},{"Start":"01:22.595 ","End":"01:25.700","Text":"but we replace top and bottom by their derivatives,"},{"Start":"01:25.700 ","End":"01:27.455","Text":"we\u0027ll get the same answer."},{"Start":"01:27.455 ","End":"01:30.665","Text":"In our case, since we have infinity over infinity,"},{"Start":"01:30.665 ","End":"01:34.310","Text":"we can take the derivative of top and bottom separately,"},{"Start":"01:34.310 ","End":"01:37.565","Text":"and notice that here the notation is as equal"},{"Start":"01:37.565 ","End":"01:40.835","Text":"by L\u0027Hopital in the infinity over infinity case,"},{"Start":"01:40.835 ","End":"01:46.445","Text":"so x squared plus 1 gives us 2x the denominator gives us 4x plus 1."},{"Start":"01:46.445 ","End":"01:51.030","Text":"At this point, we can actually substitute x equals infinity,"},{"Start":"01:51.030 ","End":"01:52.560","Text":"it\u0027s not really substitution,"},{"Start":"01:52.560 ","End":"01:55.415","Text":"it\u0027s more taking the limit as x goes to infinity,"},{"Start":"01:55.415 ","End":"01:57.725","Text":"the answer is actually 2 over 4."},{"Start":"01:57.725 ","End":"01:58.940","Text":"Several ways of getting to it,"},{"Start":"01:58.940 ","End":"02:02.180","Text":"one is the rule of thumb that you just take the highest power in each of"},{"Start":"02:02.180 ","End":"02:05.525","Text":"these polynomial 2x over 4x is like 2 over 4."},{"Start":"02:05.525 ","End":"02:08.480","Text":"Or we could do it with the long way of dividing top and"},{"Start":"02:08.480 ","End":"02:12.230","Text":"bottom by x and getting 2 over 4 plus 1 over x,"},{"Start":"02:12.230 ","End":"02:13.360","Text":"the 2 and the 4 stay,"},{"Start":"02:13.360 ","End":"02:16.580","Text":"but the 1 over x goes to 0 as x goes to infinity."},{"Start":"02:16.580 ","End":"02:19.385","Text":"In either case, we get 2 over 4,"},{"Start":"02:19.385 ","End":"02:23.915","Text":"which is nicer when it\u0027s simplified to 1/2, that\u0027s our answer."},{"Start":"02:23.915 ","End":"02:26.210","Text":"The next one I\u0027m going to tell you it\u0027s also going to be"},{"Start":"02:26.210 ","End":"02:29.975","Text":"an infinity over infinity L\u0027Hopital kind of a question."},{"Start":"02:29.975 ","End":"02:33.950","Text":"X goes to infinity and e^x also goes to infinity."},{"Start":"02:33.950 ","End":"02:36.200","Text":"In fact, it goes even faster to infinity,"},{"Start":"02:36.200 ","End":"02:40.330","Text":"but e to a power of a very large number is a very large number,"},{"Start":"02:40.330 ","End":"02:42.435","Text":"we get infinity over infinity."},{"Start":"02:42.435 ","End":"02:46.150","Text":"We use L\u0027Hopital\u0027s rule for infinity over infinity,"},{"Start":"02:46.150 ","End":"02:52.370","Text":"and we just differentiate the top e^x is just e^x and x is 1,"},{"Start":"02:52.370 ","End":"02:56.870","Text":"so basically we just have the limit as x goes to infinity of e^x,"},{"Start":"02:56.870 ","End":"02:59.120","Text":"and like I said before when x goes to infinity,"},{"Start":"02:59.120 ","End":"03:01.760","Text":"e^x also goes to infinity."},{"Start":"03:01.760 ","End":"03:04.480","Text":"That\u0027s our answer, infinity."}],"ID":8459},{"Watched":false,"Name":"Exercises 27-28","Duration":"3m 17s","ChapterTopicVideoID":8289,"CourseChapterTopicPlaylistID":52079,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8289.jpeg","UploadDate":"2017-03-02T01:08:10.8530000","DurationForVideoObject":"PT3M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.370","Text":"Here we have to compute this limit as x goes to infinity of this expression."},{"Start":"00:05.370 ","End":"00:08.265","Text":"First thing we tried to do is substitution."},{"Start":"00:08.265 ","End":"00:10.845","Text":"When I say substitute I really mean find the limit,"},{"Start":"00:10.845 ","End":"00:13.500","Text":"we just say substitute infinity."},{"Start":"00:13.500 ","End":"00:19.170","Text":"The natural log of x when x goes to infinity is known to be infinity,"},{"Start":"00:19.170 ","End":"00:22.020","Text":"and the x also goes to infinity."},{"Start":"00:22.020 ","End":"00:26.325","Text":"Infinity plus infinity plus 1 is infinity and e to the x is infinity."},{"Start":"00:26.325 ","End":"00:29.190","Text":"Which means that we have a justification for using"},{"Start":"00:29.190 ","End":"00:32.555","Text":"L\u0027Hopital\u0027 in the infinity over infinity case."},{"Start":"00:32.555 ","End":"00:35.150","Text":"What he said was don\u0027t compute this limit,"},{"Start":"00:35.150 ","End":"00:39.680","Text":"compute a different one which is obtained by differentiating top and bottom."},{"Start":"00:39.680 ","End":"00:43.580","Text":"Natural log of x gives us 1 over x. X gives us 1,"},{"Start":"00:43.580 ","End":"00:45.020","Text":"1 doesn\u0027t give us anything."},{"Start":"00:45.020 ","End":"00:46.775","Text":"Can e_x is e_x."},{"Start":"00:46.775 ","End":"00:53.225","Text":"At this point, what we actually do is substitute x equals infinity in shorthand notation,"},{"Start":"00:53.225 ","End":"00:56.030","Text":"e to the infinity is infinity,"},{"Start":"00:56.030 ","End":"00:59.475","Text":"and certainly 1 over infinity is 0,"},{"Start":"00:59.475 ","End":"01:00.690","Text":"and 1 is 1."},{"Start":"01:00.690 ","End":"01:03.905","Text":"Basically what we get is 1 over infinity."},{"Start":"01:03.905 ","End":"01:05.390","Text":"Again, like we said before,"},{"Start":"01:05.390 ","End":"01:07.430","Text":"1 over infinity is 0."},{"Start":"01:07.430 ","End":"01:09.155","Text":"That\u0027s the answer to this one."},{"Start":"01:09.155 ","End":"01:10.955","Text":"In this next exercise,"},{"Start":"01:10.955 ","End":"01:14.480","Text":"we have limit as x goes to infinity of this expression."},{"Start":"01:14.480 ","End":"01:17.210","Text":"First thing we do is try and substitute."},{"Start":"01:17.210 ","End":"01:20.945","Text":"What we get is infinity in the denominator."},{"Start":"01:20.945 ","End":"01:24.265","Text":"The limit of this is infinity."},{"Start":"01:24.265 ","End":"01:25.860","Text":"Infinity squared is infinity,"},{"Start":"01:25.860 ","End":"01:28.259","Text":"plus twice infinity is still infinity,"},{"Start":"01:28.259 ","End":"01:31.460","Text":"and a constant doesn\u0027t have any impact on infinity,"},{"Start":"01:31.460 ","End":"01:33.560","Text":"so we have infinity over infinity,"},{"Start":"01:33.560 ","End":"01:36.020","Text":"and that gives us justification to use"},{"Start":"01:36.020 ","End":"01:39.695","Text":"L\u0027Hopital\u0027s rule in the case of infinity over infinity,"},{"Start":"01:39.695 ","End":"01:41.435","Text":"which means that instead of this limit,"},{"Start":"01:41.435 ","End":"01:42.920","Text":"we compute a different limit,"},{"Start":"01:42.920 ","End":"01:46.355","Text":"which is the derivative of the top over the derivative of the bottom."},{"Start":"01:46.355 ","End":"01:47.405","Text":"Now at the top,"},{"Start":"01:47.405 ","End":"01:50.630","Text":"use the chain rule as a function of x, but it\u0027s squared."},{"Start":"01:50.630 ","End":"01:52.740","Text":"We first do the outer derivative,"},{"Start":"01:52.740 ","End":"01:55.560","Text":"something squared is twice that something and the"},{"Start":"01:55.560 ","End":"01:58.530","Text":"internal derivative tells us to indicate by prime."},{"Start":"01:58.530 ","End":"02:01.605","Text":"Natural log of x is 1 over x joins the 2,"},{"Start":"02:01.605 ","End":"02:04.470","Text":"constant gives nothing and x gives 1."},{"Start":"02:04.470 ","End":"02:06.045","Text":"This is what we have."},{"Start":"02:06.045 ","End":"02:08.345","Text":"Now just to complete the derivation,"},{"Start":"02:08.345 ","End":"02:09.620","Text":"we have this prime here."},{"Start":"02:09.620 ","End":"02:14.195","Text":"Just replaced natural log of x by 1 over x."},{"Start":"02:14.195 ","End":"02:17.030","Text":"Here\u0027s the same thing just with this thing derived."},{"Start":"02:17.030 ","End":"02:20.194","Text":"What we can do here is a bit of algebra."},{"Start":"02:20.194 ","End":"02:22.265","Text":"Notice that there\u0027s an x in the denominator here."},{"Start":"02:22.265 ","End":"02:25.400","Text":"Last thing to do would be to take 1 over x outside the brackets,"},{"Start":"02:25.400 ","End":"02:28.175","Text":"and then the x goes into the denominator."},{"Start":"02:28.175 ","End":"02:32.150","Text":"We\u0027re just left with twice natural log of x plus 2."},{"Start":"02:32.150 ","End":"02:33.650","Text":"Again, x goes to infinity,"},{"Start":"02:33.650 ","End":"02:36.875","Text":"so we try substituting and this is infinity."},{"Start":"02:36.875 ","End":"02:38.520","Text":"This we already know is infinity."},{"Start":"02:38.520 ","End":"02:43.160","Text":"Twice infinity is infinity plus 2 is still infinity over infinity."},{"Start":"02:43.160 ","End":"02:45.710","Text":"Now infinity over infinity means that\u0027s"},{"Start":"02:45.710 ","End":"02:48.835","Text":"L\u0027Hopital again with the infinity over infinity case,"},{"Start":"02:48.835 ","End":"02:50.145","Text":"which I\u0027ve written here."},{"Start":"02:50.145 ","End":"02:51.829","Text":"What we do is L\u0027Hopital,"},{"Start":"02:51.829 ","End":"02:53.330","Text":"it means we differentiate."},{"Start":"02:53.330 ","End":"02:55.655","Text":"At the top 2 goes to nothing."},{"Start":"02:55.655 ","End":"02:57.530","Text":"Natural log of x is 1 over x,"},{"Start":"02:57.530 ","End":"02:59.045","Text":"but the 2 sticks to it."},{"Start":"02:59.045 ","End":"03:00.410","Text":"The x gives us 1."},{"Start":"03:00.410 ","End":"03:06.170","Text":"Basically, we have twice 1 over x and forget the 1, the 2 stays."},{"Start":"03:06.170 ","End":"03:08.120","Text":"We just substitute x equals infinity,"},{"Start":"03:08.120 ","End":"03:10.130","Text":"we get twice 1 over infinity."},{"Start":"03:10.130 ","End":"03:11.660","Text":"Now what\u0027s 1 over infinity?"},{"Start":"03:11.660 ","End":"03:13.370","Text":"Everyone knows that\u0027s 0."},{"Start":"03:13.370 ","End":"03:15.455","Text":"Twice 0 is 0,"},{"Start":"03:15.455 ","End":"03:18.030","Text":"and that\u0027s the answer."}],"ID":8460},{"Watched":false,"Name":"Exercises 29","Duration":"3m 10s","ChapterTopicVideoID":8290,"CourseChapterTopicPlaylistID":52079,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8290.jpeg","UploadDate":"2017-01-25T01:43:38.8300000","DurationForVideoObject":"PT3M10S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.585","Text":"Here we have to compute the limit as x goes to infinity of the hyperbolic tangent of x."},{"Start":"00:06.585 ","End":"00:09.750","Text":"The first thing to try is a straightforward substitution."},{"Start":"00:09.750 ","End":"00:12.090","Text":"When I say substitute x equals infinity,"},{"Start":"00:12.090 ","End":"00:15.195","Text":"I really mean find the limit as x goes to infinity."},{"Start":"00:15.195 ","End":"00:21.615","Text":"It\u0027s not clear as yet that this really is an infinity over infinity or 0 over 0."},{"Start":"00:21.615 ","End":"00:22.755","Text":"We have to first of all,"},{"Start":"00:22.755 ","End":"00:28.185","Text":"at least start a regular means of computing what this thing is and say,"},{"Start":"00:28.185 ","End":"00:32.880","Text":"the tangent hyperbolic is sine over the cosine hyperbolic."},{"Start":"00:32.880 ","End":"00:37.310","Text":"The sine hyperbolic is what\u0027s written on the top but divided by 2,"},{"Start":"00:37.310 ","End":"00:40.830","Text":"and the cosine hyperbolic is exactly what\u0027s written on the bottom,"},{"Start":"00:40.830 ","End":"00:42.140","Text":"but divided by 2."},{"Start":"00:42.140 ","End":"00:44.515","Text":"The over 2 here and over 2 here,"},{"Start":"00:44.515 ","End":"00:45.765","Text":"cancel each other out."},{"Start":"00:45.765 ","End":"00:50.480","Text":"Let\u0027s see if we can manipulate it algebraically until we can see what to do."},{"Start":"00:50.480 ","End":"00:55.100","Text":"At this point already it is clear though that it is infinity over infinity,"},{"Start":"00:55.100 ","End":"01:01.160","Text":"because this limit we know is infinity and e to the minus x is just 1 over e to the x,"},{"Start":"01:01.160 ","End":"01:02.539","Text":"that\u0027s 1 over infinity,"},{"Start":"01:02.539 ","End":"01:04.760","Text":"so that\u0027s 0, and this is 0 also."},{"Start":"01:04.760 ","End":"01:08.930","Text":"We have infinity minus 0 over infinity plus 0,"},{"Start":"01:08.930 ","End":"01:13.660","Text":"so it\u0027s infinity over infinity and so I\u0027m going to use L\u0027Hopital."},{"Start":"01:13.660 ","End":"01:16.955","Text":"The thing is, I\u0027m not going to use L\u0027Hopital right away."},{"Start":"01:16.955 ","End":"01:19.685","Text":"Because if I did it right away, what I would get,"},{"Start":"01:19.685 ","End":"01:24.530","Text":"you can easily see would be the same thing except here with a plus and here with a minus."},{"Start":"01:24.530 ","End":"01:26.810","Text":"If at it again, it would go so back and forth."},{"Start":"01:26.810 ","End":"01:30.500","Text":"What we should do is do some algebraic manipulation so"},{"Start":"01:30.500 ","End":"01:34.370","Text":"we don\u0027t get 1 of these impossible and repetitive situations."},{"Start":"01:34.370 ","End":"01:37.670","Text":"1 suggestion is to put a common denominator"},{"Start":"01:37.670 ","End":"01:41.285","Text":"for the top and the bottom and then cancel the common denominators."},{"Start":"01:41.285 ","End":"01:45.210","Text":"Well, e to the minus x is 1 over e to the x."},{"Start":"01:45.210 ","End":"01:47.675","Text":"It\u0027s clear that the common denominator at the top"},{"Start":"01:47.675 ","End":"01:50.420","Text":"will be e to the x and similarly at the bottom."},{"Start":"01:50.420 ","End":"01:53.900","Text":"In fact, I\u0027m going to get the same thing just 1 with a minus and 1 with the plus."},{"Start":"01:53.900 ","End":"01:57.500","Text":"What we do get is after we do that is this."},{"Start":"01:57.500 ","End":"02:01.565","Text":"The next step is just to throw out the e to the x."},{"Start":"02:01.565 ","End":"02:06.065","Text":"It\u0027s on the top and on the bottom and we end up with getting the limit"},{"Start":"02:06.065 ","End":"02:10.955","Text":"of e to the 2x minus 1 over e to the 2x plus 1."},{"Start":"02:10.955 ","End":"02:13.415","Text":"This is just the algebraic simplification."},{"Start":"02:13.415 ","End":"02:16.295","Text":"But if we do apply L\u0027Hopital to it,"},{"Start":"02:16.295 ","End":"02:18.290","Text":"then it would just come out differently."},{"Start":"02:18.290 ","End":"02:19.550","Text":"There\u0027s nothing wrong with this."},{"Start":"02:19.550 ","End":"02:23.060","Text":"It\u0027s just that we haven\u0027t done any actual differentiation yet."},{"Start":"02:23.060 ","End":"02:24.635","Text":"Just as a reminder,"},{"Start":"02:24.635 ","End":"02:25.805","Text":"from the chain rule,"},{"Start":"02:25.805 ","End":"02:28.460","Text":"we have e to the power of a function of x."},{"Start":"02:28.460 ","End":"02:30.035","Text":"When we differentiate it,"},{"Start":"02:30.035 ","End":"02:32.565","Text":"we have to remember the internal derivative."},{"Start":"02:32.565 ","End":"02:33.820","Text":"What we get is,"},{"Start":"02:33.820 ","End":"02:37.190","Text":"e to the 2x would normally give us just e to the 2x but we"},{"Start":"02:37.190 ","End":"02:41.105","Text":"have to remember the internal derivative of 2x which is 2."},{"Start":"02:41.105 ","End":"02:43.445","Text":"Similarly at the bottom we get the same thing."},{"Start":"02:43.445 ","End":"02:46.850","Text":"The minus 1 disappears and so it does here."},{"Start":"02:46.850 ","End":"02:50.570","Text":"Technically this bit here really should be put"},{"Start":"02:50.570 ","End":"02:54.290","Text":"in front of here because this is where we\u0027ve actually done the differentiation."},{"Start":"02:54.290 ","End":"02:56.630","Text":"But that\u0027s of no consequence."},{"Start":"02:56.630 ","End":"02:59.480","Text":"Continuing, we just see that what we have at"},{"Start":"02:59.480 ","End":"03:01.925","Text":"the top and what we have at the bottom are the same thing."},{"Start":"03:01.925 ","End":"03:04.520","Text":"Whatever x is, this over this is 1."},{"Start":"03:04.520 ","End":"03:07.310","Text":"We have the limit of a constant function 1,"},{"Start":"03:07.310 ","End":"03:11.430","Text":"and that is just 1. That\u0027s the answer."}],"ID":8461},{"Watched":false,"Name":"Exercises 30","Duration":"2m 43s","ChapterTopicVideoID":8291,"CourseChapterTopicPlaylistID":52079,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8291.jpeg","UploadDate":"2017-01-25T01:44:22.2870000","DurationForVideoObject":"PT2M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.360","Text":"In this exercise, we have to compute the following limit."},{"Start":"00:03.360 ","End":"00:05.819","Text":"Notice that x goes to 0 plus,"},{"Start":"00:05.819 ","End":"00:08.955","Text":"which means x goes to 0 from the right."},{"Start":"00:08.955 ","End":"00:12.885","Text":"It\u0027s like x is 0.000001."},{"Start":"00:12.885 ","End":"00:15.285","Text":"It\u0027s bigger than 0 but it approaches 0."},{"Start":"00:15.285 ","End":"00:17.790","Text":"In fact, it wouldn\u0027t even make sense to have x approach"},{"Start":"00:17.790 ","End":"00:20.420","Text":"0 from the left because of the natural log."},{"Start":"00:20.420 ","End":"00:23.765","Text":"Natural log is defined only positive numbers."},{"Start":"00:23.765 ","End":"00:25.850","Text":"It\u0027s not defined on 0 or negative."},{"Start":"00:25.850 ","End":"00:27.380","Text":"This is the only way it could be."},{"Start":"00:27.380 ","End":"00:31.235","Text":"We also know that when x approaches 0 from the right,"},{"Start":"00:31.235 ","End":"00:34.834","Text":"at the limit of the natural log, is minus infinity."},{"Start":"00:34.834 ","End":"00:39.320","Text":"The only thing missing in this piece of logic is that when x goes to 0 from the right,"},{"Start":"00:39.320 ","End":"00:41.810","Text":"sine x also goes to 0 from the right."},{"Start":"00:41.810 ","End":"00:44.240","Text":"In other words, if x is positive and very, very small,"},{"Start":"00:44.240 ","End":"00:46.220","Text":"then sine x is also positive and very,"},{"Start":"00:46.220 ","End":"00:48.215","Text":"very small, and the same for the tangent."},{"Start":"00:48.215 ","End":"00:52.250","Text":"In this case, we have basically minus infinity over minus infinity."},{"Start":"00:52.250 ","End":"00:53.490","Text":"Or if we throw in a minus,"},{"Start":"00:53.490 ","End":"00:55.335","Text":"it could be infinity over infinity,"},{"Start":"00:55.335 ","End":"00:58.810","Text":"in the event, this is the case for L\u0027Hopital."},{"Start":"00:59.060 ","End":"01:02.240","Text":"L\u0027Hopital is the same thing for infinity over"},{"Start":"01:02.240 ","End":"01:05.615","Text":"infinity because it really is the minuses where we do cancel."},{"Start":"01:05.615 ","End":"01:07.760","Text":"We take this new limit,"},{"Start":"01:07.760 ","End":"01:11.720","Text":"which is obtained by differentiating both top and bottom."},{"Start":"01:11.720 ","End":"01:13.685","Text":"In this case, because of the natural log,"},{"Start":"01:13.685 ","End":"01:15.080","Text":"we get 1 over."},{"Start":"01:15.080 ","End":"01:17.225","Text":"That\u0027s the outer derivative times the inner,"},{"Start":"01:17.225 ","End":"01:20.630","Text":"which is the derivative of sine x and here likewise,"},{"Start":"01:20.630 ","End":"01:22.530","Text":"but cotangent of x."},{"Start":"01:22.530 ","End":"01:29.450","Text":"Sine x derivative is cosine x and tangent x derivative is 1 over cosine squared x."},{"Start":"01:29.450 ","End":"01:31.880","Text":"That takes us up to here."},{"Start":"01:31.880 ","End":"01:34.790","Text":"What we\u0027re going to do here is a bit of algebra."},{"Start":"01:34.790 ","End":"01:36.800","Text":"We don\u0027t want the tangent,"},{"Start":"01:36.800 ","End":"01:39.185","Text":"we wanted all in terms of sine and cosine,"},{"Start":"01:39.185 ","End":"01:42.155","Text":"so the tangent is sine over cosine."},{"Start":"01:42.155 ","End":"01:44.285","Text":"If we move things around,"},{"Start":"01:44.285 ","End":"01:45.890","Text":"then it\u0027s in the bottom,"},{"Start":"01:45.890 ","End":"01:48.965","Text":"this cosine cancels with the cosine squared."},{"Start":"01:48.965 ","End":"01:52.590","Text":"So it\u0027s just 1 over sine times 1 over cosine."},{"Start":"01:52.590 ","End":"01:54.040","Text":"What I\u0027m saying is that here,"},{"Start":"01:54.040 ","End":"01:59.405","Text":"dividing by sine over cosine is like multiplying by cosine over sine and then cancels."},{"Start":"01:59.405 ","End":"02:04.280","Text":"From here, we have the same quantity in the numerator and denominator."},{"Start":"02:04.280 ","End":"02:06.845","Text":"This 1 over sine x can be canceled."},{"Start":"02:06.845 ","End":"02:11.360","Text":"Notice that it\u0027s defined because when x is close to 0 but not 0,"},{"Start":"02:11.360 ","End":"02:13.100","Text":"then sine x can\u0027t be 0."},{"Start":"02:13.100 ","End":"02:15.500","Text":"The next time it\u0027s 0 is high,"},{"Start":"02:15.500 ","End":"02:18.245","Text":"over 180 degrees and we\u0027re close to 0."},{"Start":"02:18.245 ","End":"02:19.490","Text":"This can be canceled."},{"Start":"02:19.490 ","End":"02:20.750","Text":"This makes sense even."},{"Start":"02:20.750 ","End":"02:25.850","Text":"Cosine is certainly not 0 when x is close to 0 because the cosine of 0 is 1."},{"Start":"02:25.850 ","End":"02:28.025","Text":"We end up with canceling with this,"},{"Start":"02:28.025 ","End":"02:29.360","Text":"multiplying by the inverse,"},{"Start":"02:29.360 ","End":"02:32.000","Text":"we get cosine squared of x has a limit."},{"Start":"02:32.000 ","End":"02:34.730","Text":"Here, it doesn\u0027t really matter whether we go from the left or the right."},{"Start":"02:34.730 ","End":"02:37.205","Text":"We can just substitute x equals 0."},{"Start":"02:37.205 ","End":"02:41.840","Text":"Cosine 0 is 1 and 1 squared is 1."},{"Start":"02:41.840 ","End":"02:44.459","Text":"That\u0027s it for this exercise."}],"ID":8462},{"Watched":false,"Name":"Exercise 31","Duration":"3m 12s","ChapterTopicVideoID":28766,"CourseChapterTopicPlaylistID":52079,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/28766.jpeg","UploadDate":"2022-08-22T08:04:21.4370000","DurationForVideoObject":"PT3M12S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.150","Text":"In this exercise, we have a real valued function f on the reals"},{"Start":"00:04.150 ","End":"00:09.885","Text":"and there\u0027s a point c where the second derivative of f exists."},{"Start":"00:09.885 ","End":"00:15.090","Text":"We have to show using L\u0027Hopital\u0027s rule that this second derivative is"},{"Start":"00:15.090 ","End":"00:20.759","Text":"given by this limit as here and it\u0027s not a part of the question."},{"Start":"00:20.759 ","End":"00:25.080","Text":"Well, we have to give an example where the above limit exists,"},{"Start":"00:25.080 ","End":"00:27.165","Text":"this limit on the left,"},{"Start":"00:27.165 ","End":"00:31.035","Text":"but that f double prime of c does not exist."},{"Start":"00:31.035 ","End":"00:34.965","Text":"Now, the second derivative of f exists."},{"Start":"00:34.965 ","End":"00:38.100","Text":"The first derivative exists in some neighborhood of"},{"Start":"00:38.100 ","End":"00:42.860","Text":"c. L\u0027hopital\u0027s rule says that compute this limit,"},{"Start":"00:42.860 ","End":"00:49.650","Text":"we can differentiate top and bottom and if this limit exists, so does this."},{"Start":"00:49.650 ","End":"00:57.200","Text":"Let\u0027s differentiate, get f prime of c plus h and the inner derivative is 1."},{"Start":"00:57.200 ","End":"01:03.230","Text":"This is a constant as far as h goes, so that\u0027s 0 and here we get f prime of c"},{"Start":"01:03.230 ","End":"01:09.920","Text":"minus h and there\u0027s a minus here because the anti-derivative is minus 1."},{"Start":"01:09.920 ","End":"01:12.215","Text":"Now we\u0027ll do a bit of algebra on this."},{"Start":"01:12.215 ","End":"01:16.475","Text":"The 2 here can be taken outside"},{"Start":"01:16.475 ","End":"01:25.310","Text":"the brackets and we can add and subtract f prime of c,"},{"Start":"01:25.310 ","End":"01:27.680","Text":"and that won\u0027t change anything."},{"Start":"01:27.680 ","End":"01:32.525","Text":"We\u0027ve now got this to be the sum of 2 limits."},{"Start":"01:32.525 ","End":"01:37.805","Text":"Now both these limits express the second derivative"},{"Start":"01:37.805 ","End":"01:44.105","Text":"of f at c. This one\u0027s on the right and this one\u0027s the derivative on the left."},{"Start":"01:44.105 ","End":"01:47.020","Text":"But it is differentiable that both the same."},{"Start":"01:47.020 ","End":"01:50.690","Text":"They\u0027re both equal to F double prime of c or the derivative of"},{"Start":"01:50.690 ","End":"01:56.510","Text":"f prime and so this comes out to be just f double prime of c,"},{"Start":"01:56.510 ","End":"01:58.280","Text":"which is what we wanted."},{"Start":"01:58.280 ","End":"02:00.140","Text":"That\u0027s the first part of the question."},{"Start":"02:00.140 ","End":"02:05.960","Text":"Now we have to give this example and the example will be the signum,"},{"Start":"02:05.960 ","End":"02:08.570","Text":"the sine function of x,"},{"Start":"02:08.570 ","End":"02:12.680","Text":"meaning 1 when x is positive minus 1 when x is negative,"},{"Start":"02:12.680 ","End":"02:18.740","Text":"and 0 when x is 0 and let\u0027s take our point C to"},{"Start":"02:18.740 ","End":"02:25.815","Text":"be the point 0 and note that F is an odd function as this symmetry."},{"Start":"02:25.815 ","End":"02:30.214","Text":"Computing this limit with c equals 0,"},{"Start":"02:30.214 ","End":"02:32.015","Text":"this is what we get."},{"Start":"02:32.015 ","End":"02:33.710","Text":"F of 0 is 0,"},{"Start":"02:33.710 ","End":"02:36.680","Text":"so we\u0027re just left with this."},{"Start":"02:36.680 ","End":"02:40.070","Text":"Since F is an odd function,"},{"Start":"02:40.070 ","End":"02:46.315","Text":"we can pull the minus here out in front and we get f of h minus f of h"},{"Start":"02:46.315 ","End":"02:54.075","Text":"and this is just 0 and the limit of 0 is 0."},{"Start":"02:54.075 ","End":"02:59.254","Text":"The limit exists and it\u0027s equal to 0."},{"Start":"02:59.254 ","End":"03:06.875","Text":"But f double prime of 0 doesn\u0027t exist because F is not even continuous at 0."},{"Start":"03:06.875 ","End":"03:12.940","Text":"This is the required example and we are done."}],"ID":30117}],"Thumbnail":null,"ID":52079},{"Name":"Zero Times Infinity","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"0·∞","Duration":"18m 43s","ChapterTopicVideoID":1461,"CourseChapterTopicPlaylistID":52080,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1461.jpeg","UploadDate":"2019-10-29T12:45:29.0000000","DurationForVideoObject":"PT18M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.410 ","End":"00:05.985","Text":"In the previous part of this 4-parts series,"},{"Start":"00:05.985 ","End":"00:10.920","Text":"we learned how L\u0027Hopital\u0027s rule can help us in the cases of limits of"},{"Start":"00:10.920 ","End":"00:17.505","Text":"the form 0 over 0 and also infinity over infinity."},{"Start":"00:17.505 ","End":"00:24.270","Text":"In this part, we\u0027ll learn how L\u0027Hopital\u0027s rule can be used also in different forms."},{"Start":"00:24.270 ","End":"00:29.805","Text":"The main example will be the form 0 times infinity."},{"Start":"00:29.805 ","End":"00:36.300","Text":"This will be done by converting the question to 1 of these 2 forms."},{"Start":"00:36.300 ","End":"00:42.215","Text":"Now I want to explain what I mean by 0 over infinity,"},{"Start":"00:42.215 ","End":"00:45.205","Text":"and then I\u0027ll explain how to convert."},{"Start":"00:45.205 ","End":"00:49.895","Text":"By the way, 0 times infinity is not the only form,"},{"Start":"00:49.895 ","End":"00:53.780","Text":"there are other things which we will see such as 1 to the power of"},{"Start":"00:53.780 ","End":"00:59.610","Text":"infinity and infinity to the power of 0 and so forth."},{"Start":"00:59.920 ","End":"01:05.225","Text":"When we say limit of the form 0 times infinity,"},{"Start":"01:05.225 ","End":"01:07.985","Text":"we mean something like this;"},{"Start":"01:07.985 ","End":"01:13.280","Text":"the limit as x goes to whatever,"},{"Start":"01:13.280 ","End":"01:16.790","Text":"let\u0027s say a of f of"},{"Start":"01:16.790 ","End":"01:24.625","Text":"x times times g of x."},{"Start":"01:24.625 ","End":"01:30.710","Text":"It turns out that if you take the limit as x goes to a of f of x,"},{"Start":"01:30.710 ","End":"01:34.605","Text":"we get a 0."},{"Start":"01:34.605 ","End":"01:36.465","Text":"When x goes to a,"},{"Start":"01:36.465 ","End":"01:40.265","Text":"g of x goes to infinity and that\u0027s what I mean."},{"Start":"01:40.265 ","End":"01:43.010","Text":"We don\u0027t actually mean 0 times infinity,"},{"Start":"01:43.010 ","End":"01:48.020","Text":"but some function which tends to 0 times some other function which tends to infinity,"},{"Start":"01:48.020 ","End":"01:50.634","Text":"we want the limit of the product."},{"Start":"01:50.634 ","End":"01:53.190","Text":"Now you might say, \"Well,"},{"Start":"01:53.190 ","End":"01:56.880","Text":"what\u0027s the problem, 0 times anything is 0?\""},{"Start":"01:56.880 ","End":"01:59.600","Text":"The answer would be just 0."},{"Start":"01:59.600 ","End":"02:02.930","Text":"On the other hand, I could say anything times infinity is infinity,"},{"Start":"02:02.930 ","End":"02:04.685","Text":"so the answer should be infinity."},{"Start":"02:04.685 ","End":"02:08.590","Text":"Well, it turns out that in the case of 0 times infinity,"},{"Start":"02:08.590 ","End":"02:12.964","Text":"in this sense is that 1 function goes to 0 and the other goes to infinity,"},{"Start":"02:12.964 ","End":"02:15.845","Text":"the product could actually go to anything you want."},{"Start":"02:15.845 ","End":"02:17.510","Text":"You could name me a number,"},{"Start":"02:17.510 ","End":"02:22.570","Text":"17 minus 24, even 0, and even infinity."},{"Start":"02:22.570 ","End":"02:28.625","Text":"I could make 0 times infinity be whatever you want, including infinity."},{"Start":"02:28.625 ","End":"02:32.220","Text":"I can give you some examples."},{"Start":"02:32.330 ","End":"02:36.520","Text":"I will do that in a moment."},{"Start":"02:37.130 ","End":"02:42.170","Text":"I\u0027ll show you how to convert a product into a quotient;"},{"Start":"02:42.170 ","End":"02:45.035","Text":"it means multiplication into a division,"},{"Start":"02:45.035 ","End":"02:50.060","Text":"because really L\u0027Hopital-like quotients something over something,"},{"Start":"02:50.060 ","End":"02:55.500","Text":"and in particular 0 over 0 or infinity times infinity."},{"Start":"02:56.240 ","End":"03:02.525","Text":"Here\u0027s some examples of why we don\u0027t know what 0 times infinity is."},{"Start":"03:02.525 ","End":"03:06.650","Text":"Perhaps it\u0027s best to go straight into the examples."},{"Start":"03:06.650 ","End":"03:13.430","Text":"In the examples, I\u0027ll also show you what I mean by converting a product into a quotient."},{"Start":"03:13.430 ","End":"03:17.990","Text":"We\u0027ll also see some possible answers for 0 times infinity."},{"Start":"03:17.990 ","End":"03:24.880","Text":"Let\u0027s take some examples."},{"Start":"03:25.970 ","End":"03:32.610","Text":"These will all be examples of 0 times infinity."},{"Start":"03:32.840 ","End":"03:38.930","Text":"Then in each case we\u0027ll convert them into a quotient and end up with 1 of these 2 forms,"},{"Start":"03:38.930 ","End":"03:42.725","Text":"which really are the essence of L\u0027Hopital\u0027s quotients,"},{"Start":"03:42.725 ","End":"03:45.305","Text":"0 over 0 or infinity over infinity."},{"Start":"03:45.305 ","End":"03:47.030","Text":"For the first example,"},{"Start":"03:47.030 ","End":"03:52.595","Text":"let\u0027s take the limit as x goes to infinity"},{"Start":"03:52.595 ","End":"04:01.960","Text":"of natural log of x times 1 over x."},{"Start":"04:02.080 ","End":"04:11.630","Text":"Now you see what I mean by infinity times 0 because natural log of infinity,"},{"Start":"04:11.630 ","End":"04:13.645","Text":"if I have just the substitute,"},{"Start":"04:13.645 ","End":"04:16.440","Text":"and I would get here infinity."},{"Start":"04:16.440 ","End":"04:22.655","Text":"Here 1 over x when x goes to infinity is 1 over infinity, which is 0."},{"Start":"04:22.655 ","End":"04:29.770","Text":"This is of the form infinity times 0."},{"Start":"04:31.250 ","End":"04:33.600","Text":"Although this is a product,"},{"Start":"04:33.600 ","End":"04:36.875","Text":"it\u0027s a forced product."},{"Start":"04:36.875 ","End":"04:39.605","Text":"It\u0027s more naturally a quotient."},{"Start":"04:39.605 ","End":"04:44.629","Text":"I could just as easily have written it as the limit"},{"Start":"04:44.629 ","End":"04:50.955","Text":"as x goes to infinity of natural log of x over x."},{"Start":"04:50.955 ","End":"04:53.525","Text":"Instead of multiplying by 1 over x,"},{"Start":"04:53.525 ","End":"04:56.135","Text":"I could be dividing by x."},{"Start":"04:56.135 ","End":"04:59.600","Text":"Then if I substituted x equals infinity,"},{"Start":"04:59.600 ","End":"05:06.305","Text":"then this would be of the form infinity over infinity,"},{"Start":"05:06.305 ","End":"05:08.360","Text":"because when x goes to infinity,"},{"Start":"05:08.360 ","End":"05:10.040","Text":"x goes to infinity,"},{"Start":"05:10.040 ","End":"05:12.545","Text":"but also natural log of x goes to infinity."},{"Start":"05:12.545 ","End":"05:17.910","Text":"At this point, I would then be able to write this"},{"Start":"05:18.110 ","End":"05:27.560","Text":"as equal to according to L\u0027Hopital in the infinity over infinity case."},{"Start":"05:27.560 ","End":"05:36.050","Text":"Write this in square brackets as the derivative of the numerator,"},{"Start":"05:36.050 ","End":"05:40.380","Text":"which is, well, first of all,"},{"Start":"05:40.380 ","End":"05:43.370","Text":"the limit I should need to write in there,"},{"Start":"05:43.370 ","End":"05:44.914","Text":"x goes to infinity."},{"Start":"05:44.914 ","End":"05:48.095","Text":"Derivative of the numerator is 1 over x,"},{"Start":"05:48.095 ","End":"05:52.600","Text":"derivative of the denominator is just 1,"},{"Start":"05:52.600 ","End":"05:54.765","Text":"so it\u0027s just 1 over x."},{"Start":"05:54.765 ","End":"05:57.530","Text":"If I put x goes to infinity,"},{"Start":"05:57.530 ","End":"06:00.755","Text":"I get 1 over infinity,"},{"Start":"06:00.755 ","End":"06:03.320","Text":"which is in fact 0."},{"Start":"06:03.320 ","End":"06:05.405","Text":"In this particular case,"},{"Start":"06:05.405 ","End":"06:12.730","Text":"0 times infinity is actually 0 in this case."},{"Start":"06:12.730 ","End":"06:17.585","Text":"I wonder if that\u0027s always true or not."},{"Start":"06:17.585 ","End":"06:20.760","Text":"Let\u0027s try another example."},{"Start":"06:27.110 ","End":"06:33.610","Text":"Here, the limit as x goes to infinity of e to the minus x times x."},{"Start":"06:33.610 ","End":"06:35.170","Text":"That\u0027s our next example."},{"Start":"06:35.170 ","End":"06:37.450","Text":"Let\u0027s see what will happen here."},{"Start":"06:37.450 ","End":"06:43.060","Text":"Again, this is a product and we can convert it to a quotient."},{"Start":"06:43.060 ","End":"06:46.675","Text":"This is 1 of those types with an exponent."},{"Start":"06:46.675 ","End":"06:51.325","Text":"From algebra, when we reverse the sign of an exponent,"},{"Start":"06:51.325 ","End":"06:56.995","Text":"the expression goes into the denominator or becomes the reciprocal."},{"Start":"06:56.995 ","End":"07:01.990","Text":"What I mean is that e to the minus x is 1 over e to the plus x."},{"Start":"07:01.990 ","End":"07:05.200","Text":"In other words, this is equal to"},{"Start":"07:05.200 ","End":"07:13.745","Text":"the limit as x goes to infinity of x over e to the x."},{"Start":"07:13.745 ","End":"07:17.275","Text":"Now this is a quotient, not a product."},{"Start":"07:17.275 ","End":"07:20.870","Text":"In fact, if we check what it amounts to,"},{"Start":"07:20.870 ","End":"07:25.405","Text":"it becomes infinity over infinity,"},{"Start":"07:25.405 ","End":"07:31.075","Text":"because e to the infinity rather is also infinity."},{"Start":"07:31.075 ","End":"07:35.290","Text":"Now we can use L\u0027Hopital."},{"Start":"07:35.290 ","End":"07:41.980","Text":"This thing, write it here,"},{"Start":"07:41.980 ","End":"07:51.730","Text":"by L\u0027Hopital in the case of infinity over infinity,"},{"Start":"07:53.450 ","End":"08:01.690","Text":"it equals the limit"},{"Start":"08:03.590 ","End":"08:10.210","Text":"of 1 over e to the x."},{"Start":"08:13.520 ","End":"08:21.210","Text":"Also, x goes to infinity by differentiating top and bottom."},{"Start":"08:21.210 ","End":"08:30.390","Text":"At this point, we can just substitute x equals infinity and get e^infinity is infinity,"},{"Start":"08:30.390 ","End":"08:32.715","Text":"so this is equal to 1"},{"Start":"08:32.715 ","End":"08:42.165","Text":"over infinity and that equals 0."},{"Start":"08:42.165 ","End":"08:44.430","Text":"It may be a coincidence."},{"Start":"08:44.430 ","End":"08:48.960","Text":"Once again we got infinity times 0, which was here,"},{"Start":"08:48.960 ","End":"08:55.680","Text":"we had the 0"},{"Start":"08:55.680 ","End":"09:01.800","Text":"here and we had the infinity here in the beginning."},{"Start":"09:01.800 ","End":"09:07.210","Text":"Once again, it appears that infinity times 0 is 0."},{"Start":"09:07.640 ","End":"09:10.170","Text":"That can\u0027t truly be,"},{"Start":"09:10.170 ","End":"09:11.280","Text":"or maybe it can be,"},{"Start":"09:11.280 ","End":"09:13.425","Text":"let\u0027s check a 3rd example."},{"Start":"09:13.425 ","End":"09:19.035","Text":"The 3rd example is, so here it is."},{"Start":"09:19.035 ","End":"09:26.925","Text":"It\u0027s the limit as x goes to 0 from the right of natural log of x times x."},{"Start":"09:26.925 ","End":"09:33.090","Text":"Let\u0027s see if we just substitute 0 plus the natural log of 0 plus is"},{"Start":"09:33.090 ","End":"09:37.320","Text":"minus infinity because the limit of x goes to 0 from the right"},{"Start":"09:37.320 ","End":"09:42.180","Text":"of natural log of x is minus infinity and here it\u0027s 0."},{"Start":"09:42.180 ","End":"09:47.445","Text":"We have a minus infinity times a 0."},{"Start":"09:47.445 ","End":"09:54.405","Text":"Well, I can just highlight those 0 times infinity."},{"Start":"09:54.405 ","End":"09:57.459","Text":"We had 1 here,"},{"Start":"09:59.720 ","End":"10:04.740","Text":"these are the 3 examples infinity times 0,"},{"Start":"10:04.740 ","End":"10:08.100","Text":"0 times infinity, minus infinity times 0."},{"Start":"10:08.100 ","End":"10:12.090","Text":"In this case, we can also turn the product into"},{"Start":"10:12.090 ","End":"10:17.445","Text":"a quotient and the easiest way to do this is to write it as,"},{"Start":"10:17.445 ","End":"10:19.784","Text":"first of all, let me copy,"},{"Start":"10:19.784 ","End":"10:22.995","Text":"limit x goes to 0 plus,"},{"Start":"10:22.995 ","End":"10:30.854","Text":"and now I\u0027ll write this product as a quotient of natural log of x divided"},{"Start":"10:30.854 ","End":"10:35.295","Text":"by 1 over x because I can take something"},{"Start":"10:35.295 ","End":"10:40.635","Text":"from the numerator and put it in the denominator as it\u0027s reciprocal."},{"Start":"10:40.635 ","End":"10:43.950","Text":"In this case, now,"},{"Start":"10:43.950 ","End":"10:48.195","Text":"the natural log of 0 plus is minus infinity as we said,"},{"Start":"10:48.195 ","End":"10:53.280","Text":"and 1 over x as x goes to 0 is infinity,"},{"Start":"10:53.280 ","End":"10:56.475","Text":"in fact, this is plus infinity because it goes from the right."},{"Start":"10:56.475 ","End":"11:05.650","Text":"What we have here is in fact the minus infinity over plus infinity."},{"Start":"11:06.950 ","End":"11:15.315","Text":"What we get is this equals, according to L\u0027Hopital,"},{"Start":"11:15.315 ","End":"11:20.230","Text":"and here we have minus infinity over plus infinity,"},{"Start":"11:22.070 ","End":"11:24.479","Text":"which is just infinity,"},{"Start":"11:24.479 ","End":"11:27.180","Text":"I emphasize it by saying plus,"},{"Start":"11:27.180 ","End":"11:29.565","Text":"to the derivative here,"},{"Start":"11:29.565 ","End":"11:33.690","Text":"which is 1 over, first of all,"},{"Start":"11:33.690 ","End":"11:42.435","Text":"let\u0027s write the limit x goes to 0 plus of the derivative here is 1 over x,"},{"Start":"11:42.435 ","End":"11:48.370","Text":"the derivative here is minus 1 over x squared."},{"Start":"11:50.330 ","End":"11:55.920","Text":"This equals, we can just simplify this divide by a fraction,"},{"Start":"11:55.920 ","End":"11:58.920","Text":"you multiply by the inverse fraction."},{"Start":"11:58.920 ","End":"12:07.800","Text":"We get the limit as x goes to"},{"Start":"12:07.800 ","End":"12:16.440","Text":"0 plus of minus x simply and put x equals 0,"},{"Start":"12:16.440 ","End":"12:20.530","Text":"we get minus 0, which is just 0."},{"Start":"12:20.630 ","End":"12:24.750","Text":"Yet again, the answer came out to be 0,"},{"Start":"12:24.750 ","End":"12:29.655","Text":"but I\u0027m still not convinced that infinity times 0 is 0."},{"Start":"12:29.655 ","End":"12:33.300","Text":"There\u0027s something else I want to point"},{"Start":"12:33.300 ","End":"12:37.635","Text":"out that when we have a product that we want to convert into a quotient,"},{"Start":"12:37.635 ","End":"12:41.235","Text":"there\u0027s always at least 2 ways of doing it."},{"Start":"12:41.235 ","End":"12:45.825","Text":"Which do we leave on the top and which do we take to the bottom?"},{"Start":"12:45.825 ","End":"12:48.465","Text":"Well, in some cases it\u0027s obvious."},{"Start":"12:48.465 ","End":"12:50.340","Text":"In this case, for example,"},{"Start":"12:50.340 ","End":"12:52.710","Text":"we had natural log of x times 1 over x."},{"Start":"12:52.710 ","End":"12:54.510","Text":"Well, this is already in quotient form,"},{"Start":"12:54.510 ","End":"12:57.855","Text":"so it seems quite natural to put that on the denominator,"},{"Start":"12:57.855 ","End":"13:00.165","Text":"but it\u0027s not a guarantee,"},{"Start":"13:00.165 ","End":"13:02.535","Text":"it could have worked better the other way."},{"Start":"13:02.535 ","End":"13:05.535","Text":"Here, we had e^minus x,"},{"Start":"13:05.535 ","End":"13:07.680","Text":"which was like 1 over e^x,"},{"Start":"13:07.680 ","End":"13:12.370","Text":"so it was natural to take this on the top and take this to the bottom."},{"Start":"13:12.370 ","End":"13:17.480","Text":"In the case of natural log of x times x, not so clear."},{"Start":"13:17.480 ","End":"13:21.590","Text":"However, I did choose to take the x to"},{"Start":"13:21.590 ","End":"13:26.175","Text":"the bottom because if I leave the natural log of x on the top,"},{"Start":"13:26.175 ","End":"13:27.525","Text":"when I derive it,"},{"Start":"13:27.525 ","End":"13:30.900","Text":"there is no more logarithm anymore and I\u0027ve just gotten rid of it."},{"Start":"13:30.900 ","End":"13:32.640","Text":"If I had left it on the bottom,"},{"Start":"13:32.640 ","End":"13:34.215","Text":"it still would have been there."},{"Start":"13:34.215 ","End":"13:36.330","Text":"Still, if you do it the wrong way around,"},{"Start":"13:36.330 ","End":"13:39.045","Text":"you can always start again and try it the other way."},{"Start":"13:39.045 ","End":"13:42.300","Text":"In general, there\u0027s something I can say that, in general,"},{"Start":"13:42.300 ","End":"13:48.450","Text":"it\u0027s true that you can always convert a product to a quotient even, let\u0027s take some,"},{"Start":"13:48.450 ","End":"13:51.660","Text":"and maybe it\u0027s silly cases or obvious,"},{"Start":"13:51.660 ","End":"13:56.070","Text":"but in arithmetic, if I have,"},{"Start":"13:56.070 ","End":"14:01.350","Text":"say 4 times 10,"},{"Start":"14:01.350 ","End":"14:03.405","Text":"then I can always say,"},{"Start":"14:03.405 ","End":"14:11.445","Text":"put the 4 in the denominator and this is equals to 10 divided by 1/4,"},{"Start":"14:11.445 ","End":"14:17.385","Text":"and I can also say that it\u0027s equal to 4 divided by 1/10."},{"Start":"14:17.385 ","End":"14:19.830","Text":"You can always reciprocal 1 of"},{"Start":"14:19.830 ","End":"14:24.720","Text":"these 2 factors in"},{"Start":"14:24.720 ","End":"14:31.245","Text":"the product and just put it on the bottom by reversing it."},{"Start":"14:31.245 ","End":"14:35.955","Text":"The other thing I wanted to mention is that these 3 examples,"},{"Start":"14:35.955 ","End":"14:39.210","Text":"if you can follow these 3 and understand them,"},{"Start":"14:39.210 ","End":"14:48.105","Text":"that\u0027s probably as complex as you can hope to get in a test or an exam."},{"Start":"14:48.105 ","End":"14:56.350","Text":"These are pretty representative of the kind of exercises that 1 can encounter."},{"Start":"14:58.700 ","End":"15:03.030","Text":"What I still owe you is an example that"},{"Start":"15:03.030 ","End":"15:10.230","Text":"0 times infinity can be just anything that you name."},{"Start":"15:10.230 ","End":"15:15.045","Text":"Let\u0027s say you like the number 17."},{"Start":"15:15.045 ","End":"15:20.205","Text":"If you want me to have 0 times infinity come out to be 17,"},{"Start":"15:20.205 ","End":"15:28.750","Text":"all I have to do is take the limit as x goes to"},{"Start":"15:30.410 ","End":"15:33.180","Text":"infinity"},{"Start":"15:33.180 ","End":"15:47.010","Text":"of 17x"},{"Start":"15:47.010 ","End":"15:49.305","Text":"times 1 over x,"},{"Start":"15:49.305 ","End":"15:55.005","Text":"then immediately you see that this part goes to infinity,"},{"Start":"15:55.005 ","End":"15:57.120","Text":"when x goes to infinity,"},{"Start":"15:57.120 ","End":"15:59.370","Text":"and this part goes to 0."},{"Start":"15:59.370 ","End":"16:05.205","Text":"We do have an infinity times 0,"},{"Start":"16:05.205 ","End":"16:10.110","Text":"but the limit is equal to,"},{"Start":"16:10.110 ","End":"16:14.640","Text":"if I multiply these 2 together I just get 17 because"},{"Start":"16:14.640 ","End":"16:19.260","Text":"the x over x cancels and the answer will be 17,"},{"Start":"16:19.260 ","End":"16:20.760","Text":"that\u0027s the actual answer."},{"Start":"16:20.760 ","End":"16:22.815","Text":"This shows that any number you give me."},{"Start":"16:22.815 ","End":"16:24.390","Text":"But even more than that,"},{"Start":"16:24.390 ","End":"16:29.595","Text":"I can show you that just like we had infinity times 0 is equal to 0,"},{"Start":"16:29.595 ","End":"16:35.440","Text":"we can even have an infinity times 0 equaling infinity."},{"Start":"16:35.440 ","End":"16:38.600","Text":"I\u0027ll give you an example of that."},{"Start":"16:38.600 ","End":"16:45.620","Text":"Perhaps a rather trivial or maybe think it\u0027s a silly exercise."},{"Start":"16:45.620 ","End":"16:50.180","Text":"But if you look at x squared times 1 over x as x goes to infinity,"},{"Start":"16:50.180 ","End":"16:54.440","Text":"then x squared goes to infinity also when x goes to"},{"Start":"16:54.440 ","End":"16:59.205","Text":"infinity and 1 over x goes to 0 and so once again,"},{"Start":"16:59.205 ","End":"17:02.520","Text":"we have 0 times infinity,"},{"Start":"17:02.520 ","End":"17:04.715","Text":"but in this case,"},{"Start":"17:04.715 ","End":"17:06.560","Text":"if we simplify it,"},{"Start":"17:06.560 ","End":"17:13.325","Text":"we get the limit as x goes to infinity of"},{"Start":"17:13.325 ","End":"17:16.610","Text":"x squared over x is just x and the limit of"},{"Start":"17:16.610 ","End":"17:21.020","Text":"x as x goes to infinity of x is just infinity."},{"Start":"17:21.020 ","End":"17:25.610","Text":"It was a coincidence that in these previous examples we got"},{"Start":"17:25.610 ","End":"17:32.640","Text":"0 and then we got 0 and up here we also got 0,"},{"Start":"17:32.640 ","End":"17:35.160","Text":"we got several times,"},{"Start":"17:35.160 ","End":"17:38.180","Text":"here we got it to be 17,"},{"Start":"17:38.180 ","End":"17:40.625","Text":"and here we got it to be infinity."},{"Start":"17:40.625 ","End":"17:43.820","Text":"In other words, you really don\u0027t know what infinity times 0 is,"},{"Start":"17:43.820 ","End":"17:49.550","Text":"it could be any number and can even be infinity."},{"Start":"17:49.550 ","End":"17:53.630","Text":"The other thing I\u0027d like to point out is that when we have"},{"Start":"17:53.630 ","End":"17:59.440","Text":"a quotient and when we talk about the infinity over infinity,"},{"Start":"17:59.440 ","End":"18:02.660","Text":"the same thing works if it\u0027s minus infinity"},{"Start":"18:02.660 ","End":"18:05.600","Text":"over infinity or infinity over minus infinity."},{"Start":"18:05.600 ","End":"18:08.975","Text":"It doesn\u0027t really matter if there\u0027s a minus floating around there somewhere."},{"Start":"18:08.975 ","End":"18:12.649","Text":"L\u0027Hopital rule works for minus infinity over infinity."},{"Start":"18:12.649 ","End":"18:21.690","Text":"In fact, we had 1 of those cases of, where was it?"},{"Start":"18:21.690 ","End":"18:28.960","Text":"It was this natural log of x was minus infinity over infinity L\u0027Hopital,"},{"Start":"18:28.960 ","End":"18:33.420","Text":"but that doesn\u0027t matter if there\u0027s a minus in there somewhere."},{"Start":"18:33.430 ","End":"18:38.270","Text":"That\u0027s about it."},{"Start":"18:38.270 ","End":"18:43.010","Text":"I think we\u0027re done for this part."}],"ID":1419},{"Watched":false,"Name":"Exercise 1-2","Duration":"3m 40s","ChapterTopicVideoID":8297,"CourseChapterTopicPlaylistID":52080,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8297.jpeg","UploadDate":"2017-01-25T02:02:24.5200000","DurationForVideoObject":"PT3M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.285","Text":"Here we have to find the following limit. X goes to 0."},{"Start":"00:03.285 ","End":"00:07.455","Text":"This expression, the first thing we do is we see what goes wrong if anything,"},{"Start":"00:07.455 ","End":"00:08.970","Text":"when we put x equals 0."},{"Start":"00:08.970 ","End":"00:10.440","Text":"Well, when x is 0,"},{"Start":"00:10.440 ","End":"00:12.840","Text":"then cosine of x is 1,"},{"Start":"00:12.840 ","End":"00:15.360","Text":"1 minus 1 is 0, so this is 0."},{"Start":"00:15.360 ","End":"00:17.130","Text":"Now what about the cotangent?"},{"Start":"00:17.130 ","End":"00:23.400","Text":"Well, cotangent of x is equal to cosine of x over sine x."},{"Start":"00:23.400 ","End":"00:26.325","Text":"Now, when x goes to 0, cosine x,"},{"Start":"00:26.325 ","End":"00:30.000","Text":"this always goes to 1 when x goes to 0,"},{"Start":"00:30.000 ","End":"00:32.445","Text":"but sine x is little bit different."},{"Start":"00:32.445 ","End":"00:35.895","Text":"When x goes to 0 from the right side,"},{"Start":"00:35.895 ","End":"00:39.105","Text":"sine x goes to positive 0,"},{"Start":"00:39.105 ","End":"00:42.840","Text":"0 plus something very close to 0 but positive."},{"Start":"00:42.840 ","End":"00:45.875","Text":"When x goes to 0 on the left side,"},{"Start":"00:45.875 ","End":"00:47.750","Text":"then we get 0 minus."},{"Start":"00:47.750 ","End":"00:51.980","Text":"Now this is important because 1 over 0 plus is infinity,"},{"Start":"00:51.980 ","End":"00:54.785","Text":"so 1 over cosine goes to 0,"},{"Start":"00:54.785 ","End":"01:00.395","Text":"but the cotangent can go either to plus infinity or to minus infinity,"},{"Start":"01:00.395 ","End":"01:03.440","Text":"depending on which side the 0 we go to."},{"Start":"01:03.440 ","End":"01:08.435","Text":"As it turns out, it won\u0027t matter plus or minus infinity and so everything will be okay."},{"Start":"01:08.435 ","End":"01:11.690","Text":"But essentially, what we have in this exercise"},{"Start":"01:11.690 ","End":"01:16.085","Text":"is 0 from here times either plus infinity or minus infinity."},{"Start":"01:16.085 ","End":"01:20.150","Text":"What we do is we want to convert 0 times infinity to either 0"},{"Start":"01:20.150 ","End":"01:24.245","Text":"over 0 or infinity over infinity because that\u0027s what L\u0027Hopital knows how to do."},{"Start":"01:24.245 ","End":"01:30.125","Text":"In this case, the easiest thing to do is to take the cotangent is 1 over tangent,"},{"Start":"01:30.125 ","End":"01:32.840","Text":"because 1 sine over cosine is cosine over sine."},{"Start":"01:32.840 ","End":"01:36.680","Text":"So if we write it as the 1 minus cosine from here,"},{"Start":"01:36.680 ","End":"01:39.660","Text":"but the cotangent to put it as 1 over tangent,"},{"Start":"01:39.660 ","End":"01:44.510","Text":"that helps a bit because then we can use L\u0027Hopital because we now get the tangent"},{"Start":"01:44.510 ","End":"01:49.970","Text":"of 0 is 0 and 1 minus cosine x is also 0 when x is 0."},{"Start":"01:49.970 ","End":"01:52.400","Text":"We get a 0 over 0,"},{"Start":"01:52.400 ","End":"01:54.710","Text":"that\u0027s a 0 over 0 L\u0027Hopital."},{"Start":"01:54.710 ","End":"01:55.910","Text":"To get a different limit,"},{"Start":"01:55.910 ","End":"01:57.395","Text":"this is not equal to this,"},{"Start":"01:57.395 ","End":"02:00.050","Text":"but his theorem was that the answers will be the same."},{"Start":"02:00.050 ","End":"02:02.210","Text":"1 minus cosine, derived,"},{"Start":"02:02.210 ","End":"02:04.820","Text":"1 goes, minus cosine is sine."},{"Start":"02:04.820 ","End":"02:09.345","Text":"The derivative of tangent is 1 over cosine squared."},{"Start":"02:09.345 ","End":"02:11.960","Text":"Now here, we don\u0027t have to do any simplification"},{"Start":"02:11.960 ","End":"02:14.705","Text":"because x equals 0 will go in very nicely."},{"Start":"02:14.705 ","End":"02:16.700","Text":"In the numerator, we get 0,"},{"Start":"02:16.700 ","End":"02:19.205","Text":"in the denominator, cosine of 0 is 1,"},{"Start":"02:19.205 ","End":"02:21.020","Text":"1 over 1 squared is 1,"},{"Start":"02:21.020 ","End":"02:23.795","Text":"and 0 over 1 is just 0."},{"Start":"02:23.795 ","End":"02:25.280","Text":"We\u0027re done with this 1."},{"Start":"02:25.280 ","End":"02:27.710","Text":"Here we have a 1-sided limit."},{"Start":"02:27.710 ","End":"02:29.870","Text":"The reason for this 1-sided limit is simply that"},{"Start":"02:29.870 ","End":"02:33.000","Text":"the domain of the natural log is the positive numbers."},{"Start":"02:33.000 ","End":"02:34.745","Text":"It only makes sense to do that."},{"Start":"02:34.745 ","End":"02:40.550","Text":"Once again, we have 0 times plus or minus infinity because when x goes to 0 is 0,"},{"Start":"02:40.550 ","End":"02:42.605","Text":"and when x goes to 0 from the right,"},{"Start":"02:42.605 ","End":"02:45.350","Text":"natural log of x goes to minus infinity."},{"Start":"02:45.350 ","End":"02:49.090","Text":"In fact, we have a 0 times minus infinity here."},{"Start":"02:49.090 ","End":"02:53.375","Text":"We\u0027d like this to be either 0 over 0 or infinity over infinity."},{"Start":"02:53.375 ","End":"02:59.105","Text":"What we\u0027ll do is we\u0027ll take the x down to the denominator where it will become 1 over x."},{"Start":"02:59.105 ","End":"03:00.694","Text":"That\u0027s still just algebra."},{"Start":"03:00.694 ","End":"03:02.540","Text":"But now, if we substitute,"},{"Start":"03:02.540 ","End":"03:04.790","Text":"the numerator is minus infinity,"},{"Start":"03:04.790 ","End":"03:07.010","Text":"the denominator is plus infinity."},{"Start":"03:07.010 ","End":"03:09.110","Text":"Remember x is very tiny but positive,"},{"Start":"03:09.110 ","End":"03:11.450","Text":"so 1 over that is huge and positive."},{"Start":"03:11.450 ","End":"03:15.935","Text":"We will now apply the minus infinity over infinity L\u0027Hopital\u0027s rule."},{"Start":"03:15.935 ","End":"03:20.000","Text":"What we needed to do was to differentiate the top is 1 over x,"},{"Start":"03:20.000 ","End":"03:23.990","Text":"differentiate the bottom, you have minus 1 over x squared."},{"Start":"03:23.990 ","End":"03:25.775","Text":"If you do a bit of algebra,"},{"Start":"03:25.775 ","End":"03:28.895","Text":"what you see you\u0027ll get is the x squared over x,"},{"Start":"03:28.895 ","End":"03:30.050","Text":"the minus is still there."},{"Start":"03:30.050 ","End":"03:32.440","Text":"In other words, minus x is what we have."},{"Start":"03:32.440 ","End":"03:33.890","Text":"When x goes to 0,"},{"Start":"03:33.890 ","End":"03:37.970","Text":"this thing will be just substitute 0 minus 0,"},{"Start":"03:37.970 ","End":"03:41.760","Text":"and minus 0 is just 0, and that\u0027s it."}],"ID":8468},{"Watched":false,"Name":"Exercise 3","Duration":"3m 31s","ChapterTopicVideoID":8298,"CourseChapterTopicPlaylistID":52080,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8298.jpeg","UploadDate":"2017-01-25T02:03:21.9470000","DurationForVideoObject":"PT3M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"Here we have to compute the 1 sided limit when"},{"Start":"00:02.640 ","End":"00:05.520","Text":"x goes to 3 from the right of this expression."},{"Start":"00:05.520 ","End":"00:07.530","Text":"The reason for the 3-plus, of course,"},{"Start":"00:07.530 ","End":"00:12.300","Text":"is because the natural logarithm whose argument has to be strictly positive."},{"Start":"00:12.300 ","End":"00:14.339","Text":"So for this to be strictly positive,"},{"Start":"00:14.339 ","End":"00:15.945","Text":"x has to be bigger than 3."},{"Start":"00:15.945 ","End":"00:19.980","Text":"Now, first thing we try to do is substitute, try x equals 3."},{"Start":"00:19.980 ","End":"00:23.235","Text":"3 squared minus 9 is 0, for is this part."},{"Start":"00:23.235 ","End":"00:26.380","Text":"The other part, 3 plus minus 3 is 0-plus,"},{"Start":"00:26.380 ","End":"00:29.475","Text":"natural log of 0-plus is minus infinity."},{"Start":"00:29.475 ","End":"00:33.694","Text":"So we get to 1 of those situations of 0 and infinity."},{"Start":"00:33.694 ","End":"00:36.815","Text":"What we want to do is make it either 0 over 0"},{"Start":"00:36.815 ","End":"00:40.400","Text":"or infinity over infinity pluses and/or minuses."},{"Start":"00:40.400 ","End":"00:42.110","Text":"To get it to this situation,"},{"Start":"00:42.110 ","End":"00:45.905","Text":"we have to put 1 of these 2 on the denominator, inverted."},{"Start":"00:45.905 ","End":"00:47.900","Text":"Turns out the right 1 to do is this 1,"},{"Start":"00:47.900 ","End":"00:49.250","Text":"this 1 goes on the bottom."},{"Start":"00:49.250 ","End":"00:50.855","Text":"When this goes on the bottom,"},{"Start":"00:50.855 ","End":"00:55.170","Text":"then this thing stays but this goes on the bottom but as a reciprocal."},{"Start":"00:55.170 ","End":"00:58.490","Text":"What we need to do now is see what we\u0027re into."},{"Start":"00:58.490 ","End":"01:01.390","Text":"This thing is, as we said, minus infinity."},{"Start":"01:01.390 ","End":"01:05.840","Text":"At the bottom, it actually turns out to be plus infinity because x"},{"Start":"01:05.840 ","End":"01:10.700","Text":"is positive and slightly bigger than 3 so x squared will be slightly bigger than 9,"},{"Start":"01:10.700 ","End":"01:13.570","Text":"which means we have 1 over something positive,"},{"Start":"01:13.570 ","End":"01:16.610","Text":"so the limit must be plus infinity."},{"Start":"01:16.610 ","End":"01:20.555","Text":"We do have a situation with minus infinity over infinity,"},{"Start":"01:20.555 ","End":"01:22.640","Text":"but it doesn\u0027t matter about the pluses and minuses."},{"Start":"01:22.640 ","End":"01:24.965","Text":"We can always use the L\u0027Hospital rule."},{"Start":"01:24.965 ","End":"01:27.690","Text":"Here what we did is we would write it like this,"},{"Start":"01:27.690 ","End":"01:32.135","Text":"L\u0027Hospital in the case of a minus infinity over infinity and then we write a new limit,"},{"Start":"01:32.135 ","End":"01:34.610","Text":"but 1 which is guaranteed to give the same answer and"},{"Start":"01:34.610 ","End":"01:37.380","Text":"that is to differentiate both top and bottom."},{"Start":"01:37.380 ","End":"01:40.475","Text":"For the top natural logarithm gives us 1 over,"},{"Start":"01:40.475 ","End":"01:43.475","Text":"you might say, what about the internal derivative, but it\u0027s 1."},{"Start":"01:43.475 ","End":"01:47.485","Text":"For the other 1, we have a chain rule for 1 over box,"},{"Start":"01:47.485 ","End":"01:49.020","Text":"and I derive it,"},{"Start":"01:49.020 ","End":"01:52.550","Text":"I get minus 1 over the box squared."},{"Start":"01:52.550 ","End":"01:55.130","Text":"But then I have to take the internal derivative,"},{"Start":"01:55.130 ","End":"01:56.705","Text":"in other words, box-prime,"},{"Start":"01:56.705 ","End":"01:58.280","Text":"whatever is here, derived."},{"Start":"01:58.280 ","End":"02:02.135","Text":"It should be minus 1 here over this thing squared,"},{"Start":"02:02.135 ","End":"02:04.520","Text":"but we also have to factor in this,"},{"Start":"02:04.520 ","End":"02:09.530","Text":"where what\u0027s in the box is x squared minus 9 and its derivative is 2x."},{"Start":"02:09.530 ","End":"02:12.680","Text":"We had the minus 1 to multiply that by 2x,"},{"Start":"02:12.680 ","End":"02:15.020","Text":"this is what we get up to here."},{"Start":"02:15.020 ","End":"02:19.865","Text":"Now continuing, we just simplify this algebraically."},{"Start":"02:19.865 ","End":"02:25.310","Text":"The 1 and we don\u0027t see on the x minus 3 is on the bottom,"},{"Start":"02:25.310 ","End":"02:26.690","Text":"but the whole denominator,"},{"Start":"02:26.690 ","End":"02:29.390","Text":"we flip it over so that the x squared minus 9"},{"Start":"02:29.390 ","End":"02:32.465","Text":"comes to the top and the minus 2x stays at the bottom."},{"Start":"02:32.465 ","End":"02:34.250","Text":"From here, we can simplify and"},{"Start":"02:34.250 ","End":"02:37.955","Text":"even factor because we do have difference of squares rule that"},{"Start":"02:37.955 ","End":"02:46.155","Text":"A squared minus B squared is equal to A minus B and its conjugate A plus B."},{"Start":"02:46.155 ","End":"02:49.815","Text":"Here I\u0027m taking the A to be x and B to be"},{"Start":"02:49.815 ","End":"02:54.090","Text":"3 so we have x squared minus 3 squared.So if we have that,"},{"Start":"02:54.090 ","End":"02:58.640","Text":"then what we get is this is x minus 3 times x plus 3,"},{"Start":"02:58.640 ","End":"02:59.810","Text":"but we have squared,"},{"Start":"02:59.810 ","End":"03:03.950","Text":"so we put the square on each 1 of the denominator as it was before."},{"Start":"03:03.950 ","End":"03:08.555","Text":"If we cancel this x minus 3 with 1 of these,"},{"Start":"03:08.555 ","End":"03:10.655","Text":"then we get this expression."},{"Start":"03:10.655 ","End":"03:14.630","Text":"Now we\u0027re okay with substituting x equals 3."},{"Start":"03:14.630 ","End":"03:16.550","Text":"But the thing is that,"},{"Start":"03:16.550 ","End":"03:18.740","Text":"well, this bottom it matters,"},{"Start":"03:18.740 ","End":"03:21.800","Text":"it\u0027s only that it\u0027s not 0, but the x minus 3,"},{"Start":"03:21.800 ","End":"03:24.350","Text":"if x goes to 3, this goes to 0."},{"Start":"03:24.350 ","End":"03:29.210","Text":"Once we found the 0 over here and the whole thing and all the others are not 0,"},{"Start":"03:29.210 ","End":"03:32.910","Text":"then it just gives us an answer of 0."}],"ID":8469},{"Watched":false,"Name":"Exercise 4","Duration":"3m 15s","ChapterTopicVideoID":8299,"CourseChapterTopicPlaylistID":52080,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8299.jpeg","UploadDate":"2017-01-25T02:04:15.9730000","DurationForVideoObject":"PT3M15S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.690","Text":"In this exercise, we have to compute the limit when x goes to"},{"Start":"00:03.690 ","End":"00:08.295","Text":"infinity of x times natural log of this fraction here."},{"Start":"00:08.295 ","End":"00:12.990","Text":"If we just naively go ahead and try and substitute x equals infinity,"},{"Start":"00:12.990 ","End":"00:15.480","Text":"which means take the limit as x goes to infinity,"},{"Start":"00:15.480 ","End":"00:18.165","Text":"then here, this x gives us infinity,"},{"Start":"00:18.165 ","End":"00:24.930","Text":"and this will just take the x plus 3 over x minus 3 and rewrite it as if I divide top and"},{"Start":"00:24.930 ","End":"00:32.985","Text":"bottom by x as this part goes to 1 plus 3 over x divided by 1 minus,"},{"Start":"00:32.985 ","End":"00:38.535","Text":"and then you can do the limit as x goes to infinity first and then take the natural log,"},{"Start":"00:38.535 ","End":"00:40.335","Text":"we can do it the other way around."},{"Start":"00:40.335 ","End":"00:44.120","Text":"What we get is as x goes to infinity,"},{"Start":"00:44.120 ","End":"00:48.620","Text":"goes to 0, this same thing also goes to 0."},{"Start":"00:48.620 ","End":"00:52.790","Text":"We get 1 plus 0 over 1 minus 0, that\u0027s just 1,"},{"Start":"00:52.790 ","End":"00:57.575","Text":"and then we take the natural log of 1 and that gives us 0."},{"Start":"00:57.575 ","End":"01:02.210","Text":"So 0 from here and the x itself goes to infinity."},{"Start":"01:02.210 ","End":"01:07.040","Text":"That means that we have a situation of infinity times 0,"},{"Start":"01:07.040 ","End":"01:11.930","Text":"and that\u0027s very close to L\u0027Hopital because this can easily be converted to either 0 over"},{"Start":"01:11.930 ","End":"01:17.615","Text":"0 or possibly infinity over infinity by doing a bit of algebraic manipulation."},{"Start":"01:17.615 ","End":"01:19.205","Text":"In this case, to trial and error,"},{"Start":"01:19.205 ","End":"01:22.850","Text":"you should try it with the x in the denominator and make it 1 over x,"},{"Start":"01:22.850 ","End":"01:25.880","Text":"and then this stays in the numerator, and now,"},{"Start":"01:25.880 ","End":"01:27.845","Text":"instead of infinity times 0,"},{"Start":"01:27.845 ","End":"01:31.535","Text":"we now have a 0 over 0 situation."},{"Start":"01:31.535 ","End":"01:35.060","Text":"It goes to 0 and this bottom also goes to 0."},{"Start":"01:35.060 ","End":"01:37.925","Text":"We have 1 of those 0 over 0 things,"},{"Start":"01:37.925 ","End":"01:42.790","Text":"and the notation to write this is a different limit than this."},{"Start":"01:42.790 ","End":"01:46.940","Text":"L\u0027Hopital was the one who demonstrated that this technique gives the same answer."},{"Start":"01:46.940 ","End":"01:49.115","Text":"It isn\u0027t the same, and this is a different limit."},{"Start":"01:49.115 ","End":"01:51.260","Text":"With 0 over 0, this thing just works."},{"Start":"01:51.260 ","End":"01:52.700","Text":"To differentiate top and bottom,"},{"Start":"01:52.700 ","End":"01:55.205","Text":"you\u0027ll get the same answer the hard way or some other way."},{"Start":"01:55.205 ","End":"01:57.590","Text":"How do we differentiate the top?"},{"Start":"01:57.590 ","End":"01:58.850","Text":"Well, it\u0027s a chain rule."},{"Start":"01:58.850 ","End":"02:00.740","Text":"It\u0027s a natural log of something."},{"Start":"02:00.740 ","End":"02:04.985","Text":"The answer is 1 over that something times the internal derivative,"},{"Start":"02:04.985 ","End":"02:08.450","Text":"by which I mean the derivative of this fraction. Now it\u0027s a fraction."},{"Start":"02:08.450 ","End":"02:12.650","Text":"So obviously, going to use the quotient rule and just remind you of it."},{"Start":"02:12.650 ","End":"02:14.705","Text":"The f over g prime,"},{"Start":"02:14.705 ","End":"02:19.805","Text":"f prime g minus fg prime all over g squared."},{"Start":"02:19.805 ","End":"02:23.390","Text":"The derivative of this following this formula will give us this,"},{"Start":"02:23.390 ","End":"02:26.865","Text":"and the denominator, 1 over x is minus 1 over x squared."},{"Start":"02:26.865 ","End":"02:28.655","Text":"Begin with x to the minus 1,"},{"Start":"02:28.655 ","End":"02:31.370","Text":"lower the power by 1 and multiply in front."},{"Start":"02:31.370 ","End":"02:35.855","Text":"Anyway, this is what we get and this is the answer for the algebraic simplification."},{"Start":"02:35.855 ","End":"02:37.850","Text":"Now, here we have a rational function,"},{"Start":"02:37.850 ","End":"02:40.070","Text":"meaning a polynomial over a polynomial."},{"Start":"02:40.070 ","End":"02:43.295","Text":"1 way to find the limit as x goes to infinity"},{"Start":"02:43.295 ","End":"02:46.580","Text":"is for each of the polynomials for the top and for the bottom,"},{"Start":"02:46.580 ","End":"02:48.485","Text":"we just take the leading term,"},{"Start":"02:48.485 ","End":"02:51.530","Text":"meaning the term with the highest power of x in it."},{"Start":"02:51.530 ","End":"02:53.150","Text":"Well, there is only 1 term here,"},{"Start":"02:53.150 ","End":"02:55.925","Text":"so we take a 6x squared and on the bottom,"},{"Start":"02:55.925 ","End":"02:58.535","Text":"we open it up, it\u0027s x squared minus 9."},{"Start":"02:58.535 ","End":"03:02.090","Text":"We just take the x squared part because that\u0027s the highest power."},{"Start":"03:02.090 ","End":"03:03.470","Text":"After you get this,"},{"Start":"03:03.470 ","End":"03:06.920","Text":"then cancel this x squared with this x squared."},{"Start":"03:06.920 ","End":"03:09.200","Text":"So it\u0027s just 6 now. It\u0027s a constant function."},{"Start":"03:09.200 ","End":"03:12.455","Text":"The limit of any constant function is just the constant."},{"Start":"03:12.455 ","End":"03:16.470","Text":"The answer is just 6 and we\u0027re done."}],"ID":8470},{"Watched":false,"Name":"Exercise 5","Duration":"3m 14s","ChapterTopicVideoID":1464,"CourseChapterTopicPlaylistID":52080,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1464.jpeg","UploadDate":"2014-10-01T10:04:59.1870000","DurationForVideoObject":"PT3M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.865","Text":"In this exercise, we have to find the following limit."},{"Start":"00:02.865 ","End":"00:04.785","Text":"When x goes to infinity,"},{"Start":"00:04.785 ","End":"00:08.175","Text":"the first thing to do is just to try substituting x equals infinity,"},{"Start":"00:08.175 ","End":"00:09.900","Text":"which really means taking the limit."},{"Start":"00:09.900 ","End":"00:13.980","Text":"For the first bit we have x going to infinity,"},{"Start":"00:13.980 ","End":"00:15.345","Text":"so this is infinity."},{"Start":"00:15.345 ","End":"00:18.090","Text":"The second bit we get 0. How do we get 0?"},{"Start":"00:18.090 ","End":"00:24.720","Text":"Well, 5 over infinity is 0,1 plus 0 is 1 and so on."},{"Start":"00:24.720 ","End":"00:36.240","Text":"You get the idea, you quickly get to infinity times 0."},{"Start":"00:36.240 ","End":"00:40.760","Text":"This usually indicates that we\u0027re close to using L\u0027Hopital\u0027s,"},{"Start":"00:40.760 ","End":"00:45.875","Text":"we can easily bring this to either infinity over infinity or 0 over 0."},{"Start":"00:45.875 ","End":"00:48.380","Text":"There after you\u0027ve had some practice,"},{"Start":"00:48.380 ","End":"00:51.740","Text":"you\u0027ll know that the one to try is for"},{"Start":"00:51.740 ","End":"00:58.015","Text":"the 0 over 0 by taking the x and putting it in the denominator."},{"Start":"00:58.015 ","End":"01:00.150","Text":"Of course, when it goes in the denominator,"},{"Start":"01:00.150 ","End":"01:01.875","Text":"it becomes 1 over x."},{"Start":"01:01.875 ","End":"01:04.985","Text":"This bit just serves as the numerator."},{"Start":"01:04.985 ","End":"01:07.675","Text":"This we already said was 0."},{"Start":"01:07.675 ","End":"01:09.630","Text":"When x goes to infinity,"},{"Start":"01:09.630 ","End":"01:11.470","Text":"1 over x is also 0."},{"Start":"01:11.470 ","End":"01:14.510","Text":"We have a 0 over 0 limit here."},{"Start":"01:14.510 ","End":"01:20.660","Text":"If we decide to use L\u0027Hopital\u0027s rule, why wouldn\u0027t we?"},{"Start":"01:20.660 ","End":"01:26.720","Text":"Then we change this limit to an equivalent limit that will give us the same answer,"},{"Start":"01:26.720 ","End":"01:28.850","Text":"but it\u0027s actually a different limit."},{"Start":"01:28.850 ","End":"01:35.340","Text":"That is the one we get by differentiating top and bottom separately."},{"Start":"01:35.770 ","End":"01:39.295","Text":"What we get is,"},{"Start":"01:39.295 ","End":"01:43.980","Text":"on the bottom, we differentiate 1 over x,"},{"Start":"01:43.980 ","End":"01:46.640","Text":"so we get minus 1 over x squared."},{"Start":"01:46.640 ","End":"01:49.340","Text":"On the top, we have a square root,"},{"Start":"01:49.340 ","End":"01:54.095","Text":"and the derivative of square root is 1 over twice the square root,"},{"Start":"01:54.095 ","End":"01:56.075","Text":"which gives us this part."},{"Start":"01:56.075 ","End":"01:58.490","Text":"But there\u0027s also the extra bit which is the"},{"Start":"01:58.490 ","End":"02:02.330","Text":"internal derivative because there\u0027s an internal function,"},{"Start":"02:02.330 ","End":"02:04.265","Text":"1 plus 5 over x."},{"Start":"02:04.265 ","End":"02:10.100","Text":"The 1 goes to nothing and 5 over x goes to minus 5 over x squared."},{"Start":"02:10.100 ","End":"02:15.390","Text":"Nothing else because the minus 1 also just goes to nothing."},{"Start":"02:15.730 ","End":"02:18.870","Text":"We have this."},{"Start":"02:20.110 ","End":"02:24.930","Text":"If we simplify it a bit,"},{"Start":"02:25.090 ","End":"02:29.225","Text":"then what we get is,"},{"Start":"02:29.225 ","End":"02:31.865","Text":"well, it\u0027s the same thing."},{"Start":"02:31.865 ","End":"02:33.530","Text":"This and this are here and here,"},{"Start":"02:33.530 ","End":"02:39.620","Text":"but the minus 1 over x squared just becomes minus x squared in the numerator."},{"Start":"02:39.620 ","End":"02:44.700","Text":"This is very good because this minus x squared,"},{"Start":"02:44.700 ","End":"02:49.335","Text":"I can cancel this minus x squared with this minus x squared,"},{"Start":"02:49.335 ","End":"02:53.564","Text":"then all that we\u0027re left with is,"},{"Start":"02:53.564 ","End":"02:57.090","Text":"from this new 2 bit it\u0027s just the 5."},{"Start":"02:57.090 ","End":"03:01.795","Text":"At this point we can substitute x equals infinity."},{"Start":"03:01.795 ","End":"03:05.719","Text":"Here we get 5 over infinity is 0,"},{"Start":"03:05.719 ","End":"03:09.245","Text":"plus 1 is 1, square root of 1 is 1, so that\u0027s 2."},{"Start":"03:09.245 ","End":"03:11.105","Text":"Then we have the 5 from here."},{"Start":"03:11.105 ","End":"03:14.730","Text":"The answer is 5 over 2."}],"ID":1423}],"Thumbnail":null,"ID":52080},{"Name":"Exponents Infinity over Zero, Zero over Zero, One over Infinity","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"∞^0, 0^0, 1^∞","Duration":"13m 59s","ChapterTopicVideoID":1470,"CourseChapterTopicPlaylistID":52081,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1470.jpeg","UploadDate":"2019-10-29T12:42:26.0130000","DurationForVideoObject":"PT13M59S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.245","Text":"In the previous parts,"},{"Start":"00:01.245 ","End":"00:07.290","Text":"we learned how to compute limits of the form 0 over 0 and infinity over infinity."},{"Start":"00:07.290 ","End":"00:10.440","Text":"These are the classic to L\u0027Hopital forms."},{"Start":"00:10.440 ","End":"00:14.370","Text":"We also learned how to compute limits of the form 0 times infinity."},{"Start":"00:14.370 ","End":"00:18.480","Text":"Today, we\u0027re going to learn how to compute 3 new forms,"},{"Start":"00:18.480 ","End":"00:22.305","Text":"all exponents infinity to the power of 0,"},{"Start":"00:22.305 ","End":"00:23.880","Text":"0 to the power of 0,"},{"Start":"00:23.880 ","End":"00:25.875","Text":"and 1 to the power of infinity."},{"Start":"00:25.875 ","End":"00:30.645","Text":"These all stem from a limit of the form 1 function to the power of another function,"},{"Start":"00:30.645 ","End":"00:33.210","Text":"f of x to the power of g of x."},{"Start":"00:33.210 ","End":"00:39.570","Text":"The first 1 means that f is going to infinity while g goes to 0."},{"Start":"00:39.570 ","End":"00:42.440","Text":"The other case, we have 1 function that goes to 0 and the"},{"Start":"00:42.440 ","End":"00:46.515","Text":"other to the power of the other function which goes also to 0."},{"Start":"00:46.515 ","End":"00:47.930","Text":"In the third case,"},{"Start":"00:47.930 ","End":"00:52.289","Text":"something goes to 1, to the power of something that goes to infinity."},{"Start":"00:52.600 ","End":"00:56.290","Text":"I\u0027d like to emphasize what I just said."},{"Start":"00:56.290 ","End":"00:57.900","Text":"Let\u0027s take, for example,"},{"Start":"00:57.900 ","End":"00:59.535","Text":"the 1 to the power of infinity."},{"Start":"00:59.535 ","End":"01:04.520","Text":"Of course, 1 times 1 times 1 times 1 or 1 to any power is going to be still 1."},{"Start":"01:04.520 ","End":"01:09.170","Text":"But 1 to the power of infinity just means some expression, in this case,"},{"Start":"01:09.170 ","End":"01:15.530","Text":"f of x which tends to 1 to the power of another expression which tends to infinity."},{"Start":"01:15.530 ","End":"01:18.920","Text":"Now, in this case, the answer could come out to be anything."},{"Start":"01:18.920 ","End":"01:21.200","Text":"It\u0027s not that I can\u0027t compute such thing,"},{"Start":"01:21.200 ","End":"01:24.025","Text":"it just might mean that the answer will come out like"},{"Start":"01:24.025 ","End":"01:27.900","Text":"14 in 1 exercise minus 100 in another exercise,"},{"Start":"01:27.900 ","End":"01:32.850","Text":"it\u0027s just indeterminate in general but in any specific exercise, we can solve it."},{"Start":"01:32.850 ","End":"01:36.995","Text":"The same goes for 0 to the power of 0 and infinity to the power of 0."},{"Start":"01:36.995 ","End":"01:40.610","Text":"This is just a shorthand way of saying something that tends to infinity,"},{"Start":"01:40.610 ","End":"01:43.525","Text":"to the power of something that tends to 0 and so forth."},{"Start":"01:43.525 ","End":"01:47.630","Text":"A very important formula that we\u0027ll be"},{"Start":"01:47.630 ","End":"01:51.875","Text":"utilizing to solve these cases is the following formula,"},{"Start":"01:51.875 ","End":"01:58.480","Text":"a to the power of b is equal e to the power of b times natural log of a."},{"Start":"01:58.480 ","End":"02:00.495","Text":"In each of 3 cases,"},{"Start":"02:00.495 ","End":"02:02.550","Text":"we\u0027ll use this formula."},{"Start":"02:02.550 ","End":"02:06.320","Text":"I\u0027ll do some examples of each of the following,"},{"Start":"02:06.320 ","End":"02:08.030","Text":"whether it\u0027s infinity to the 0,"},{"Start":"02:08.030 ","End":"02:10.505","Text":"0 to the 0, 1 to the infinity."},{"Start":"02:10.505 ","End":"02:14.310","Text":"Whenever we have an expression to the power of an expression that\u0027s 1 of these forms,"},{"Start":"02:14.310 ","End":"02:19.655","Text":"we\u0027re going to be using this formula from algebra."},{"Start":"02:19.655 ","End":"02:22.415","Text":"On to the examples."},{"Start":"02:22.415 ","End":"02:28.550","Text":"The first example will be the limit as x"},{"Start":"02:28.550 ","End":"02:34.879","Text":"goes to infinity of x to the power of 1 over x."},{"Start":"02:34.879 ","End":"02:36.020","Text":"Now if you look at this,"},{"Start":"02:36.020 ","End":"02:39.380","Text":"this is of the form infinity to the power of 0,"},{"Start":"02:39.380 ","End":"02:41.195","Text":"because when x goes to infinity,"},{"Start":"02:41.195 ","End":"02:45.590","Text":"then x goes to infinity and 1 over x goes to 0,"},{"Start":"02:45.590 ","End":"02:47.120","Text":"so this is of the form,"},{"Start":"02:47.120 ","End":"02:51.095","Text":"I\u0027ll just write it symbolically as infinity to the 0,"},{"Start":"02:51.095 ","End":"02:52.730","Text":"not really an equals,"},{"Start":"02:52.730 ","End":"02:54.575","Text":"but that\u0027s the form it\u0027s of."},{"Start":"02:54.575 ","End":"02:59.839","Text":"I will use the formula here and rewrite this expression."},{"Start":"02:59.839 ","End":"03:02.080","Text":"This is the limit, still,"},{"Start":"03:02.080 ","End":"03:06.770","Text":"x goes to infinity of e to the power of"},{"Start":"03:06.770 ","End":"03:13.415","Text":"b is the 1 over x and natural log of a is natural log of x."},{"Start":"03:13.415 ","End":"03:20.435","Text":"Now what we\u0027re going to do for this limit is just let the top bit be asterisk."},{"Start":"03:20.435 ","End":"03:21.890","Text":"Let\u0027s say it\u0027s e to the power."},{"Start":"03:21.890 ","End":"03:28.005","Text":"Let\u0027s just say that was some asterisk or asterisk is all of this,"},{"Start":"03:28.005 ","End":"03:30.695","Text":"compute the asterisk and then at the end,"},{"Start":"03:30.695 ","End":"03:33.545","Text":"take e to the power of that answer."},{"Start":"03:33.545 ","End":"03:36.845","Text":"What we need, and this is the asterisk part,"},{"Start":"03:36.845 ","End":"03:44.255","Text":"is the limit as x goes to infinity of 1 over x natural log of x."},{"Start":"03:44.255 ","End":"03:48.680","Text":"Now if you look at it, this is of the form 0 times infinity"},{"Start":"03:48.680 ","End":"03:53.930","Text":"because 1 over infinity is 0 and the natural log of infinity is infinity."},{"Start":"03:53.930 ","End":"03:55.610","Text":"We\u0027ve actually done this before."},{"Start":"03:55.610 ","End":"03:59.630","Text":"It\u0027s 1 of those where we have a product and we easily turn it into a quotient,"},{"Start":"03:59.630 ","End":"04:03.980","Text":"it almost begs us the 1 over x really wants us to look at it"},{"Start":"04:03.980 ","End":"04:11.015","Text":"as limit of natural log of x over x as this quotient."},{"Start":"04:11.015 ","End":"04:13.250","Text":"Still, x goes to infinity."},{"Start":"04:13.250 ","End":"04:17.090","Text":"Now we have a regular L\u0027Hopital of infinity over infinity."},{"Start":"04:17.090 ","End":"04:20.245","Text":"Now we can say this equals biL\u0027Hopital,"},{"Start":"04:20.245 ","End":"04:25.890","Text":"in the infinity over infinity case to"},{"Start":"04:25.890 ","End":"04:33.635","Text":"the limit as x goes to infinity and then we derived the top and the bottom separately."},{"Start":"04:33.635 ","End":"04:36.755","Text":"We have 1 over x over 1,"},{"Start":"04:36.755 ","End":"04:38.210","Text":"and we\u0027ve done this 1 before,"},{"Start":"04:38.210 ","End":"04:41.540","Text":"1 over infinity is 0,"},{"Start":"04:41.540 ","End":"04:44.480","Text":"0 over 1 is 0."},{"Start":"04:44.480 ","End":"04:48.875","Text":"Now that\u0027s not the entire end because this is just the asterisks,"},{"Start":"04:48.875 ","End":"04:53.600","Text":"so we have to put this 0 instead of the asterisk over here."},{"Start":"04:53.600 ","End":"05:00.815","Text":"We get this is equal to e to the power of 0, which equals 1."},{"Start":"05:00.815 ","End":"05:04.340","Text":"In this case, infinity to the power of 0 is 1,"},{"Start":"05:04.340 ","End":"05:07.294","Text":"but don\u0027t assume that it works that way in other cases,"},{"Start":"05:07.294 ","End":"05:11.185","Text":"this is the indeterminate and we have to do each case separately."},{"Start":"05:11.185 ","End":"05:16.420","Text":"That was number 1, and so on to number 2."},{"Start":"05:17.630 ","End":"05:28.790","Text":"Here, we\u0027ll take the limit as x goes to 0 from the right of x to the power of x."},{"Start":"05:28.790 ","End":"05:37.154","Text":"Now, in this case, we have the other class that we had it before,"},{"Start":"05:37.154 ","End":"05:39.105","Text":"we have 1 of this kind,"},{"Start":"05:39.105 ","End":"05:42.080","Text":"0 to the power of 0 and the last example,"},{"Start":"05:42.080 ","End":"05:44.000","Text":"of course, will be 1 to the power of infinity."},{"Start":"05:44.000 ","End":"05:46.010","Text":"We\u0027ll take 1 example of each."},{"Start":"05:46.010 ","End":"05:48.220","Text":"Here, we have 0,"},{"Start":"05:48.220 ","End":"05:51.020","Text":"let\u0027s just write it as 0 to the power of 0,"},{"Start":"05:51.020 ","End":"05:52.520","Text":"I don\u0027t mean this is really equal,"},{"Start":"05:52.520 ","End":"05:53.690","Text":"I\u0027m just saying symbolically,"},{"Start":"05:53.690 ","End":"05:55.460","Text":"this is of this type."},{"Start":"05:55.460 ","End":"06:00.605","Text":"In this case, we\u0027re going to apply the formula that we have"},{"Start":"06:00.605 ","End":"06:05.570","Text":"here and that is equal to the limit as x goes"},{"Start":"06:05.570 ","End":"06:15.680","Text":"to 0 from the right of e to the power of b is x and a is also x,"},{"Start":"06:15.680 ","End":"06:19.940","Text":"so it\u0027s x natural log of x."},{"Start":"06:19.940 ","End":"06:25.100","Text":"Now what I\u0027m going to do here is take this expression"},{"Start":"06:25.100 ","End":"06:30.800","Text":"here that e to the power of and denoted as asterisk."},{"Start":"06:30.800 ","End":"06:35.990","Text":"Then we\u0027ll do the asterisk separately and then the limit"},{"Start":"06:35.990 ","End":"06:40.690","Text":"of e to the something is just e to the power of the answer of what that something is."},{"Start":"06:40.690 ","End":"06:45.530","Text":"Well, I first will figure out the limit of this and then I will put it here."},{"Start":"06:45.530 ","End":"06:47.120","Text":"We\u0027ll return to this."},{"Start":"06:47.120 ","End":"06:52.740","Text":"I\u0027ll put equal to remind me to come back here and this asterisk,"},{"Start":"06:52.940 ","End":"06:56.620","Text":"or rather the limit of the asterisk."},{"Start":"06:57.240 ","End":"07:03.970","Text":"Limit as x goes to 0"},{"Start":"07:03.970 ","End":"07:14.335","Text":"plus x times natural log of x,"},{"Start":"07:14.335 ","End":"07:17.050","Text":"this is equal to,"},{"Start":"07:17.050 ","End":"07:22.045","Text":"now here we have a situation where we have a 0 times minus infinity,"},{"Start":"07:22.045 ","End":"07:23.800","Text":"because x goes to 0,"},{"Start":"07:23.800 ","End":"07:26.950","Text":"this goes to 0 and when x goes to 0,"},{"Start":"07:26.950 ","End":"07:30.415","Text":"from the right, natural log of x goes to minus infinity."},{"Start":"07:30.415 ","End":"07:35.085","Text":"I\u0027m just writing symbolically that we have a 0 times minus infinity."},{"Start":"07:35.085 ","End":"07:40.605","Text":"We learned that what we do in this case is we turn the multiplication into a division."},{"Start":"07:40.605 ","End":"07:49.250","Text":"In this case, we take natural log of x and divide it by 1 over x."},{"Start":"07:49.250 ","End":"07:52.180","Text":"Putting x into the bottom."},{"Start":"07:52.180 ","End":"07:55.855","Text":"I forgot to write the limit here, just a second."},{"Start":"07:55.855 ","End":"08:01.330","Text":"Now if you see here we have 1 over x,"},{"Start":"08:01.330 ","End":"08:04.195","Text":"which is 1 over 0 plus, this is infinity."},{"Start":"08:04.195 ","End":"08:06.595","Text":"Here we have minus infinity."},{"Start":"08:06.595 ","End":"08:11.980","Text":"We have a case using L\u0027Hopital and"},{"Start":"08:11.980 ","End":"08:18.385","Text":"the minus infinity over infinity case."},{"Start":"08:18.385 ","End":"08:27.790","Text":"Here we differentiate, so derivative of this is 1 over x."},{"Start":"08:27.790 ","End":"08:32.080","Text":"Here, we have minus 1 over x squared."},{"Start":"08:32.080 ","End":"08:40.000","Text":"Altogether what we get is multiply by the inverse of this."},{"Start":"08:40.000 ","End":"08:43.600","Text":"Basically we get the limit as just minus x."},{"Start":"08:43.600 ","End":"08:48.175","Text":"Limit of minus x as x goes to 0,"},{"Start":"08:48.175 ","End":"08:50.035","Text":"it doesn\u0027t matter from the right."},{"Start":"08:50.035 ","End":"08:56.870","Text":"Then this is just equal to 0 minus 0, whatever."},{"Start":"08:57.000 ","End":"09:00.295","Text":"Now this was the asterisk."},{"Start":"09:00.295 ","End":"09:04.135","Text":"This whole thing was the asterisk."},{"Start":"09:04.135 ","End":"09:09.535","Text":"The asterisk means that now we put this back into here."},{"Start":"09:09.535 ","End":"09:11.935","Text":"Just putting this 0 into here."},{"Start":"09:11.935 ","End":"09:14.815","Text":"This is e to the 0,"},{"Start":"09:14.815 ","End":"09:18.410","Text":"and this is equal to 1."},{"Start":"09:19.350 ","End":"09:22.960","Text":"Here we had infinity to the 0 is 1,"},{"Start":"09:22.960 ","End":"09:24.400","Text":"but that\u0027s not going to be always true."},{"Start":"09:24.400 ","End":"09:27.040","Text":"Here we had 0 to the power of 0 is 1,"},{"Start":"09:27.040 ","End":"09:29.230","Text":"but that\u0027s not always going to be true."},{"Start":"09:29.230 ","End":"09:33.440","Text":"Let\u0027s look at number 3."},{"Start":"09:33.440 ","End":"09:36.700","Text":"Here we have number 3."},{"Start":"09:37.340 ","End":"09:40.329","Text":"This is of the third variety,"},{"Start":"09:40.329 ","End":"09:42.205","Text":"the 1 to the power of infinity."},{"Start":"09:42.205 ","End":"09:45.700","Text":"We have the limit as x goes to 0 from the right,"},{"Start":"09:45.700 ","End":"09:48.745","Text":"1 plus x to the power of 1 over x."},{"Start":"09:48.745 ","End":"09:56.304","Text":"If you look at it, 1 plus x is our 1 and 1 over x is infinity."},{"Start":"09:56.304 ","End":"09:57.790","Text":"I\u0027m just writing symbolically,"},{"Start":"09:57.790 ","End":"10:00.280","Text":"this is 1 to the power of infinity."},{"Start":"10:00.280 ","End":"10:02.380","Text":"Now we need to remember the formula."},{"Start":"10:02.380 ","End":"10:04.270","Text":"It\u0027s scrolled off the screen,"},{"Start":"10:04.270 ","End":"10:05.965","Text":"so I\u0027ll write it here at the side;"},{"Start":"10:05.965 ","End":"10:08.950","Text":"that a to the power of"},{"Start":"10:08.950 ","End":"10:16.480","Text":"b is e to the power of b times natural log of a."},{"Start":"10:16.480 ","End":"10:18.655","Text":"That\u0027s just the formula."},{"Start":"10:18.655 ","End":"10:21.220","Text":"I hope something scrolled somewhere."},{"Start":"10:21.220 ","End":"10:23.125","Text":"Anyway, back to here."},{"Start":"10:23.125 ","End":"10:27.190","Text":"We\u0027re going to use this formula to convert this to"},{"Start":"10:27.190 ","End":"10:37.225","Text":"the limit as x goes to 0 from the right of e to the power of,"},{"Start":"10:37.225 ","End":"10:41.455","Text":"now the a in this formula b is this,"},{"Start":"10:41.455 ","End":"10:50.690","Text":"which is the 1 over x and natural log of 1 plus x,"},{"Start":"10:50.850 ","End":"10:53.425","Text":"after using this formula."},{"Start":"10:53.425 ","End":"10:55.600","Text":"Now this trick you\u0027re starting to get used to"},{"Start":"10:55.600 ","End":"10:58.540","Text":"is when we have e to the power of something and the limit,"},{"Start":"10:58.540 ","End":"11:04.285","Text":"we take this exponent and just call it something say asterisk,"},{"Start":"11:04.285 ","End":"11:10.615","Text":"and we compute that separately and then take e to the power of this limit."},{"Start":"11:10.615 ","End":"11:14.650","Text":"This is equal to the power of asterisk,"},{"Start":"11:14.650 ","End":"11:18.670","Text":"which equals, now we\u0027ll see what the asterisk is."},{"Start":"11:18.670 ","End":"11:26.925","Text":"The asterisk is the limit as"},{"Start":"11:26.925 ","End":"11:36.800","Text":"x goes to 0 from the right of 1 over x,"},{"Start":"11:36.800 ","End":"11:41.200","Text":"natural log of 1 plus x."},{"Start":"11:41.200 ","End":"11:43.990","Text":"Again, as x goes to 0 from the right."},{"Start":"11:43.990 ","End":"11:50.470","Text":"Now 1 over x goes to infinity and natural log of 1 plus x,"},{"Start":"11:50.470 ","End":"11:54.445","Text":"1 plus x goes to 1 when x goes to 0."},{"Start":"11:54.445 ","End":"11:56.800","Text":"Natural log of 1 is 0."},{"Start":"11:56.800 ","End":"12:01.165","Text":"We have here infinity times 0."},{"Start":"12:01.165 ","End":"12:03.220","Text":"I\u0027ll just write it, this is just a symbolic;"},{"Start":"12:03.220 ","End":"12:05.110","Text":"isn\u0027t equal to infinity times 0."},{"Start":"12:05.110 ","End":"12:07.330","Text":"I\u0027m just saying this is variety."},{"Start":"12:07.330 ","End":"12:10.090","Text":"In this case we know what to do;"},{"Start":"12:10.090 ","End":"12:13.630","Text":"we turn the multiplication into a division,"},{"Start":"12:13.630 ","End":"12:22.105","Text":"so we get the limit as x goes to 0 from the right of"},{"Start":"12:22.105 ","End":"12:28.240","Text":"natural log of 1 plus x divided"},{"Start":"12:28.240 ","End":"12:34.600","Text":"by x."},{"Start":"12:34.600 ","End":"12:40.720","Text":"Here we have a case of L\u0027Hopital because both numerator and denominator go to 0."},{"Start":"12:40.720 ","End":"12:48.710","Text":"We have a L\u0027Hopital in the case of 0 over 0."},{"Start":"12:50.550 ","End":"12:55.480","Text":"What we do is just derive top and bottom."},{"Start":"12:55.480 ","End":"12:58.450","Text":"The bottom derives to 1,"},{"Start":"12:58.450 ","End":"13:03.410","Text":"the top derives to 1 over 1 plus x."},{"Start":"13:05.280 ","End":"13:10.270","Text":"This comes out, we can just substitute x equals 0,"},{"Start":"13:10.270 ","End":"13:12.580","Text":"we get 1 over 1 plus 0."},{"Start":"13:12.580 ","End":"13:16.360","Text":"Basically, we just get this equals 1."},{"Start":"13:16.360 ","End":"13:19.390","Text":"Now that\u0027s just the asterisk."},{"Start":"13:19.390 ","End":"13:22.660","Text":"Now I have to put this 1 which is the asterisk from"},{"Start":"13:22.660 ","End":"13:26.290","Text":"here and substitute it so it\u0027s e to the 1,"},{"Start":"13:26.290 ","End":"13:29.560","Text":"so the answer is e. In this case,"},{"Start":"13:29.560 ","End":"13:30.760","Text":"but only in this case,"},{"Start":"13:30.760 ","End":"13:33.910","Text":"1 to the infinity came out to be e. We\u0027re given"},{"Start":"13:33.910 ","End":"13:38.260","Text":"an example of 1 of each of the 3 types and the technique we use,"},{"Start":"13:38.260 ","End":"13:40.915","Text":"mainly using this formula,"},{"Start":"13:40.915 ","End":"13:42.250","Text":"which is very important,"},{"Start":"13:42.250 ","End":"13:43.615","Text":"just got buried there."},{"Start":"13:43.615 ","End":"13:48.430","Text":"Then usually taking the exponent as an asterisk,"},{"Start":"13:48.430 ","End":"13:54.145","Text":"computing it at the side and then putting e to the power of it and getting the answer."},{"Start":"13:54.145 ","End":"13:59.270","Text":"That\u0027s it for Part 3, next Part 4."}],"ID":1447},{"Watched":false,"Name":"Exercise 1","Duration":"3m 14s","ChapterTopicVideoID":8300,"CourseChapterTopicPlaylistID":52081,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8300.jpeg","UploadDate":"2017-01-25T02:21:53.8600000","DurationForVideoObject":"PT3M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.655","Text":"Here we have to compute the limit as x goes to 1 of x to the power of 1 over x minus 1."},{"Start":"00:05.655 ","End":"00:08.025","Text":"Now you might wonder why this is in red and green."},{"Start":"00:08.025 ","End":"00:09.990","Text":"Well, if we look ahead, we are going to be using"},{"Start":"00:09.990 ","End":"00:13.635","Text":"a formula that is in red and green and it will apply here."},{"Start":"00:13.635 ","End":"00:15.870","Text":"If we try to substitute x equals 1,"},{"Start":"00:15.870 ","End":"00:17.655","Text":"this is what we first usually try to do."},{"Start":"00:17.655 ","End":"00:20.415","Text":"We get 1 to the power of 1/0."},{"Start":"00:20.415 ","End":"00:23.325","Text":"Now 1/0 is either plus or minus infinity,"},{"Start":"00:23.325 ","End":"00:26.715","Text":"depending on which way we go to 1 from the left or from the right."},{"Start":"00:26.715 ","End":"00:30.300","Text":"In any event it is 1 to the infinity in a manner of speaking,"},{"Start":"00:30.300 ","End":"00:32.100","Text":"plus or minus doesn\u0027t make a big difference."},{"Start":"00:32.100 ","End":"00:35.505","Text":"The technique we use in this case is to use the following formula."},{"Start":"00:35.505 ","End":"00:38.280","Text":"This works in general for A positive."},{"Start":"00:38.280 ","End":"00:41.420","Text":"A to the power of B is equal to e to the power of B,"},{"Start":"00:41.420 ","End":"00:42.700","Text":"natural log of A."},{"Start":"00:42.700 ","End":"00:45.500","Text":"If we apply it to our limit here,"},{"Start":"00:45.500 ","End":"00:50.340","Text":"then we get that the x part is the A and the 1 over x minus 1 is the B."},{"Start":"00:50.340 ","End":"00:51.965","Text":"We get, using this formula,"},{"Start":"00:51.965 ","End":"00:56.570","Text":"e to the power of 1 over x minus 1 times natural log x as x goes to 1."},{"Start":"00:56.570 ","End":"00:59.060","Text":"Now in anticipation of L\u0027Hopital,"},{"Start":"00:59.060 ","End":"01:00.590","Text":"I have the notion that we are going to have to"},{"Start":"01:00.590 ","End":"01:02.600","Text":"figure out the limit of the exponent because,"},{"Start":"01:02.600 ","End":"01:05.570","Text":"and I have experience in this and this is of the form 1 over"},{"Start":"01:05.570 ","End":"01:09.380","Text":"x minus 1 is infinity and this part is 0."},{"Start":"01:09.380 ","End":"01:12.655","Text":"Infinity times 0 is not as good as a"},{"Start":"01:12.655 ","End":"01:17.600","Text":"0/0 so it\u0027s better to write it as a quotient and not as a product."},{"Start":"01:17.600 ","End":"01:21.040","Text":"What I mean is, just write the x minus 1 under the natural log of the x."},{"Start":"01:21.040 ","End":"01:25.985","Text":"We get a quotient and then it will go to 0/0 and we\u0027ll be able to use L\u0027Hopital."},{"Start":"01:25.985 ","End":"01:28.115","Text":"In fact, what I\u0027m going to do here,"},{"Start":"01:28.115 ","End":"01:32.030","Text":"and this is a common trick for whenever we have the e to the power of something,"},{"Start":"01:32.030 ","End":"01:35.585","Text":"and then there\u0027s a limit is to write it as e to the power of asterisk,"},{"Start":"01:35.585 ","End":"01:37.400","Text":"where asterisk is the limit of"},{"Start":"01:37.400 ","End":"01:41.090","Text":"just the exponent part without the e. We first compute the limit,"},{"Start":"01:41.090 ","End":"01:43.970","Text":"we\u0027ll just throw out the e and when we get the answer to this,"},{"Start":"01:43.970 ","End":"01:45.110","Text":"we\u0027ll finally get a number."},{"Start":"01:45.110 ","End":"01:48.230","Text":"We\u0027ll plug the number in place of the asterisk and we\u0027ll"},{"Start":"01:48.230 ","End":"01:51.995","Text":"have our answer as e to the power of the answer to the other exercise."},{"Start":"01:51.995 ","End":"01:55.340","Text":"Now it turns out that and I\u0027m looking into the future,"},{"Start":"01:55.340 ","End":"01:59.885","Text":"we get that this limit of natural log of x/x turns out to be 1."},{"Start":"01:59.885 ","End":"02:02.285","Text":"I\u0027ll show you afterwards how we got to that 1."},{"Start":"02:02.285 ","End":"02:03.665","Text":"I\u0027m just looking into the future,"},{"Start":"02:03.665 ","End":"02:07.250","Text":"asterisk comes out to be 1 so I substitute e to the power"},{"Start":"02:07.250 ","End":"02:10.910","Text":"of 1 and the answer is e. What I really did was pause here,"},{"Start":"02:10.910 ","End":"02:13.940","Text":"do a side exercise of just this part,"},{"Start":"02:13.940 ","End":"02:17.910","Text":"the asterisk, the limit as natural log of x over x minus 1."},{"Start":"02:17.910 ","End":"02:22.240","Text":"If we do this and the answer comes out 1 and then we go back here and we plug it in."},{"Start":"02:22.240 ","End":"02:24.700","Text":"That\u0027s how it works, I\u0027ve just gotten a bit ahead of myself."},{"Start":"02:24.700 ","End":"02:27.020","Text":"Now, this is equal to,"},{"Start":"02:27.020 ","End":"02:29.240","Text":"if you see that, this is 0/0,"},{"Start":"02:29.240 ","End":"02:31.445","Text":"of course, because when x goes to 1,"},{"Start":"02:31.445 ","End":"02:34.025","Text":"natural log is just 0 in any event."},{"Start":"02:34.025 ","End":"02:37.370","Text":"When x goes to 1, 1 minus 1 goes to 0."},{"Start":"02:37.370 ","End":"02:39.800","Text":"In fact, we actually, if you substitute x equals 1,"},{"Start":"02:39.800 ","End":"02:43.415","Text":"we get 0/0, but x doesn\u0027t equal 1 it only goes to 1."},{"Start":"02:43.415 ","End":"02:44.885","Text":"We use L\u0027Hopital here."},{"Start":"02:44.885 ","End":"02:48.830","Text":"Write equals L\u0027Hopital 0/0 and instead of this limit,"},{"Start":"02:48.830 ","End":"02:50.225","Text":"we replace it with a new limit,"},{"Start":"02:50.225 ","End":"02:53.705","Text":"which is what we get when we differentiate both top and bottom separately."},{"Start":"02:53.705 ","End":"02:56.180","Text":"Here it is, natural log of x goes to 1/x,"},{"Start":"02:56.180 ","End":"02:58.235","Text":"x minus 1 gives us 1."},{"Start":"02:58.235 ","End":"03:01.205","Text":"If we just substitute x equals 1 here, the answer is 1."},{"Start":"03:01.205 ","End":"03:03.110","Text":"When we get to this 1 normally,"},{"Start":"03:03.110 ","End":"03:04.745","Text":"then we go back here,"},{"Start":"03:04.745 ","End":"03:06.440","Text":"which we\u0027ll leave blank for the moment,"},{"Start":"03:06.440 ","End":"03:08.809","Text":"is e to the asterisk and then we substitute"},{"Start":"03:08.809 ","End":"03:11.560","Text":"1 instead of the asterisk and then get the answer."},{"Start":"03:11.560 ","End":"03:15.420","Text":"It\u0027s a bit back and forward but anyway, that\u0027s our answer."}],"ID":8471},{"Watched":false,"Name":"Exercise 2","Duration":"3m 25s","ChapterTopicVideoID":8301,"CourseChapterTopicPlaylistID":52081,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8301.jpeg","UploadDate":"2017-01-25T02:22:47.4300000","DurationForVideoObject":"PT3M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.970","Text":"In this exercise, you have to find the limit as x goes"},{"Start":"00:02.970 ","End":"00:06.195","Text":"to 0 from the right of ax to the power of x."},{"Start":"00:06.195 ","End":"00:09.930","Text":"It\u0027s important that we require that a be positive because"},{"Start":"00:09.930 ","End":"00:13.890","Text":"only positive numbers should be used as a base for an exponential function."},{"Start":"00:13.890 ","End":"00:15.885","Text":"We\u0027re going to move on to the solution."},{"Start":"00:15.885 ","End":"00:17.385","Text":"Here\u0027s the original question."},{"Start":"00:17.385 ","End":"00:19.455","Text":"Why is it in red and green, you might wonder?"},{"Start":"00:19.455 ","End":"00:22.785","Text":"Well, in a moment, we\u0027re about to use a formula which is in red and green."},{"Start":"00:22.785 ","End":"00:27.435","Text":"In any event, what we have if we substitute x equals 0 plus is,"},{"Start":"00:27.435 ","End":"00:32.895","Text":"this is positive number times 0 plus is still 0 plus to the power of 0,"},{"Start":"00:32.895 ","End":"00:34.650","Text":"or just 0^0."},{"Start":"00:34.650 ","End":"00:36.930","Text":"This is 1 of these indeterminate forms"},{"Start":"00:36.930 ","End":"00:40.020","Text":"and we don\u0027t know what the answer is to 0^0,"},{"Start":"00:40.020 ","End":"00:43.790","Text":"so we try to convert it into another form and where we can use L\u0027Hopital."},{"Start":"00:43.790 ","End":"00:47.130","Text":"Now in almost all these exponential forms like 0^0,"},{"Start":"00:47.130 ","End":"00:48.500","Text":"1 to the infinity,"},{"Start":"00:48.500 ","End":"00:50.075","Text":"infinity to the 0, and so on,"},{"Start":"00:50.075 ","End":"00:52.175","Text":"we almost always use the following formula,"},{"Start":"00:52.175 ","End":"00:53.735","Text":"which is that when we have an exponent,"},{"Start":"00:53.735 ","End":"00:58.180","Text":"you can always make it as an exponent with base e by using this formula here,"},{"Start":"00:58.180 ","End":"01:00.770","Text":"and in our case, if we apply it over here,"},{"Start":"01:00.770 ","End":"01:03.980","Text":"then this part is the A part and this is the B part."},{"Start":"01:03.980 ","End":"01:07.920","Text":"Any way, if you just plug it in here, we get x natural log of ax."},{"Start":"01:07.920 ","End":"01:12.590","Text":"The trick here is to compute a much simpler limit is just to get rid of"},{"Start":"01:12.590 ","End":"01:18.015","Text":"the e and to call this and to compute a new limit which I\u0027ll call asterisk,"},{"Start":"01:18.015 ","End":"01:23.645","Text":"and asterisk will be compute the limit as x goes to 0 plus of this thing,"},{"Start":"01:23.645 ","End":"01:28.760","Text":"but without the e, just off the x natural log of ax."},{"Start":"01:28.760 ","End":"01:30.290","Text":"I\u0027m going to do this separately,"},{"Start":"01:30.290 ","End":"01:34.280","Text":"and then the answer to this limit will be e to the power"},{"Start":"01:34.280 ","End":"01:39.040","Text":"of the asterisk I get when I finally solve it, and then I\u0027ll plug it back in here."},{"Start":"01:39.040 ","End":"01:41.825","Text":"Let\u0027s move on to the asterisk part,"},{"Start":"01:41.825 ","End":"01:48.755","Text":"and if we see here that this is 0 times natural log of 0 which is minus infinity,"},{"Start":"01:48.755 ","End":"01:51.050","Text":"so here computing the limit I wrote here,"},{"Start":"01:51.050 ","End":"01:53.165","Text":"we get a 0 times minus infinity,"},{"Start":"01:53.165 ","End":"01:54.875","Text":"same thing as 0 times infinity."},{"Start":"01:54.875 ","End":"01:57.380","Text":"What we do is we put 1 of these 2 things into"},{"Start":"01:57.380 ","End":"02:00.200","Text":"the denominator either the x or the natural log,"},{"Start":"02:00.200 ","End":"02:03.455","Text":"and obviously, the easiest thing to do is to put the x into denominator."},{"Start":"02:03.455 ","End":"02:05.645","Text":"We\u0027ve seen this trick many times before,"},{"Start":"02:05.645 ","End":"02:07.880","Text":"that when you throw something into the denominator,"},{"Start":"02:07.880 ","End":"02:10.050","Text":"x becomes the reciprocal 1 over."},{"Start":"02:10.050 ","End":"02:14.930","Text":"Now, we have the minus infinity over infinity because when x goes to"},{"Start":"02:14.930 ","End":"02:17.390","Text":"0 plus, this is natural log of 0 plus which is"},{"Start":"02:17.390 ","End":"02:20.825","Text":"minus infinity, and 1 over 0 plus is plus infinity."},{"Start":"02:20.825 ","End":"02:22.835","Text":"Here, we can use L\u0027Hopital,"},{"Start":"02:22.835 ","End":"02:25.140","Text":"for the minus infinity over infinity case,"},{"Start":"02:25.140 ","End":"02:29.570","Text":"and what L\u0027Hopital\u0027s Rule says is that instead of the original limit,"},{"Start":"02:29.570 ","End":"02:31.145","Text":"replace it with a different limit,"},{"Start":"02:31.145 ","End":"02:35.430","Text":"that is the 1 obtained by differentiating both top and bottom separately."},{"Start":"02:35.430 ","End":"02:38.180","Text":"As for the top, we get 1 over ax,"},{"Start":"02:38.180 ","End":"02:43.144","Text":"but times the internal derivative which is a, and for this we get minus 1 over x squared,"},{"Start":"02:43.144 ","End":"02:47.190","Text":"and actually a lot of it\u0027s simplifies because the a cancels with the a,"},{"Start":"02:47.190 ","End":"02:49.190","Text":"and when the x squared goes into the top,"},{"Start":"02:49.190 ","End":"02:52.595","Text":"x squared over x is just x and we\u0027re left with a minus."},{"Start":"02:52.595 ","End":"02:55.250","Text":"In other words, we just have to figure out the limit as x"},{"Start":"02:55.250 ","End":"02:58.490","Text":"goes to 0 of minus x, and that\u0027s just 0."},{"Start":"02:58.490 ","End":"03:04.835","Text":"Now, this 0 have to remember with the side exercise over here to compute this bit here."},{"Start":"03:04.835 ","End":"03:07.460","Text":"Now that we have that this asterisk is a 0,"},{"Start":"03:07.460 ","End":"03:11.770","Text":"we can now substitute the asterisk as e^0"},{"Start":"03:11.770 ","End":"03:14.840","Text":"by borrowing the asterisk was equal to 0,"},{"Start":"03:14.840 ","End":"03:17.945","Text":"and this is just equal to 1 and 1 is the answer."},{"Start":"03:17.945 ","End":"03:19.820","Text":"This stuff which is the side exercise is"},{"Start":"03:19.820 ","End":"03:21.665","Text":"a very common trick. Pull out the e,"},{"Start":"03:21.665 ","End":"03:26.250","Text":"compute the limit, and then the e back in. We\u0027re done."}],"ID":8472},{"Watched":false,"Name":"Exercise 3","Duration":"3m 23s","ChapterTopicVideoID":8302,"CourseChapterTopicPlaylistID":52081,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8302.jpeg","UploadDate":"2017-01-25T02:23:44.0970000","DurationForVideoObject":"PT3M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.069","Text":"Here we have to find the limit as x goes to 2 from the right of this expression,"},{"Start":"00:05.069 ","End":"00:06.645","Text":"something to the power of something."},{"Start":"00:06.645 ","End":"00:08.775","Text":"If we try substituting x equals 2,"},{"Start":"00:08.775 ","End":"00:12.705","Text":"we get twice 2 minus 4 is 0 and 2 minus 2 is 0."},{"Start":"00:12.705 ","End":"00:15.927","Text":"In other words, this is 1 of those 0^0 forms"},{"Start":"00:15.927 ","End":"00:19.410","Text":"which can usually be adapted to L\u0027Hopital."},{"Start":"00:19.410 ","End":"00:22.140","Text":"The reason that this is in red and green is because"},{"Start":"00:22.140 ","End":"00:25.035","Text":"I\u0027m going to use this formula, and in that formula,"},{"Start":"00:25.035 ","End":"00:27.900","Text":"the red here corresponds to the red here and the green to the green."},{"Start":"00:27.900 ","End":"00:31.065","Text":"If I use this formula and we\u0027ve seen this several times,"},{"Start":"00:31.065 ","End":"00:32.460","Text":"then what I get here,"},{"Start":"00:32.460 ","End":"00:35.685","Text":"instead of this is the limit, x goes to the same thing,"},{"Start":"00:35.685 ","End":"00:39.210","Text":"but e to the power of this x minus 2 from here,"},{"Start":"00:39.210 ","End":"00:42.005","Text":"log of 2x minus 4 from here."},{"Start":"00:42.005 ","End":"00:43.955","Text":"Now our usual trick is,"},{"Start":"00:43.955 ","End":"00:47.450","Text":"first we compute a separate side exercise."},{"Start":"00:47.450 ","End":"00:52.835","Text":"The limit without the e limit x goes to 2 from the right of"},{"Start":"00:52.835 ","End":"01:00.725","Text":"just the x minus 2 times natural log of 2x minus 4, and we call this asterisk."},{"Start":"01:00.725 ","End":"01:02.795","Text":"Then when we\u0027ve done computing the asterisk,"},{"Start":"01:02.795 ","End":"01:05.900","Text":"restore the e and we write e to the power of"},{"Start":"01:05.900 ","End":"01:08.224","Text":"asterisk which we found from the side exercise."},{"Start":"01:08.224 ","End":"01:09.430","Text":"This gave us some number,"},{"Start":"01:09.430 ","End":"01:10.570","Text":"you would put it back here,"},{"Start":"01:10.570 ","End":"01:12.350","Text":"e to the power of it is the answer."},{"Start":"01:12.350 ","End":"01:16.189","Text":"I\u0027ll continue by writing this limit and this is going to be our side exercise."},{"Start":"01:16.189 ","End":"01:20.000","Text":"We examined this, we always try substituting first and see what we get."},{"Start":"01:20.000 ","End":"01:22.655","Text":"We get x goes to 2 from the right."},{"Start":"01:22.655 ","End":"01:26.960","Text":"Here, 2 from the right minus 2 is 0 plus or just 0."},{"Start":"01:26.960 ","End":"01:31.640","Text":"The natural log, this also comes out to be 0 plus because we have twice 2 plus,"},{"Start":"01:31.640 ","End":"01:33.335","Text":"which is 4 plus minus 4,"},{"Start":"01:33.335 ","End":"01:34.640","Text":"that gives us 0 plus."},{"Start":"01:34.640 ","End":"01:38.045","Text":"This part is infinity and this part is just 0."},{"Start":"01:38.045 ","End":"01:42.230","Text":"What we get is that\u0027s minus infinity, the actual log of 0 plus."},{"Start":"01:42.230 ","End":"01:44.675","Text":"We get a case of 0 times minus infinity,"},{"Start":"01:44.675 ","End":"01:47.765","Text":"which is very good at 0 times infinity, same thing."},{"Start":"01:47.765 ","End":"01:51.110","Text":"What we do is convert it into 0 over 0 or infinity over"},{"Start":"01:51.110 ","End":"01:54.620","Text":"infinity by putting 1 of the 2 factors in to the denominator."},{"Start":"01:54.620 ","End":"01:56.960","Text":"Now, I\u0027d rather have x minus 2 somehow in"},{"Start":"01:56.960 ","End":"01:59.750","Text":"the denominator because then when I differentiate the numerator,"},{"Start":"01:59.750 ","End":"02:01.310","Text":"the natural log will disappear."},{"Start":"02:01.310 ","End":"02:04.640","Text":"That\u0027s how we\u0027ll do it. We\u0027ve seen this many time, I\u0027m not going to go into detail."},{"Start":"02:04.640 ","End":"02:06.080","Text":"What\u0027s in the numerator can go into"},{"Start":"02:06.080 ","End":"02:09.465","Text":"the denominator as long as you invert it or put its reciprocal."},{"Start":"02:09.465 ","End":"02:12.480","Text":"This is then the L\u0027Hopital."},{"Start":"02:12.480 ","End":"02:15.305","Text":"Here, we had 0 times minus infinity."},{"Start":"02:15.305 ","End":"02:18.170","Text":"We left the minus infinity part here and we put the"},{"Start":"02:18.170 ","End":"02:21.170","Text":"0 into the denominator which makes it a plus infinity."},{"Start":"02:21.170 ","End":"02:24.740","Text":"I over 2 plus minus 2 is 1 over 0 plus its infinity,"},{"Start":"02:24.740 ","End":"02:29.030","Text":"and so we use the L\u0027Hopital rule with the minus infinity over infinity case"},{"Start":"02:29.030 ","End":"02:33.440","Text":"and this is what we get when we differentiate natural log of 2x minus 4,"},{"Start":"02:33.440 ","End":"02:36.725","Text":"it\u0027s 1 over 2x minus 4 times the internal derivative."},{"Start":"02:36.725 ","End":"02:40.445","Text":"Over here, it\u0027s minus 1 over 1 over x minus 2 squared."},{"Start":"02:40.445 ","End":"02:42.620","Text":"There is no internal derivative or rather there is,"},{"Start":"02:42.620 ","End":"02:44.960","Text":"but it\u0027s 1 so that makes no difference."},{"Start":"02:44.960 ","End":"02:46.775","Text":"Now, we are at this point."},{"Start":"02:46.775 ","End":"02:50.900","Text":"At this point, do a bit of algebra like throw the x minus 2 squared to the top,"},{"Start":"02:50.900 ","End":"02:55.820","Text":"put the minus in front, and put the 2x minus 4 in the bottom and the 2 at the side."},{"Start":"02:55.820 ","End":"02:58.920","Text":"Anyway, after we do some algebra, 2 cancels with 2,"},{"Start":"02:58.920 ","End":"03:01.850","Text":"x minus 2 cancels 1 of these, and we\u0027re just left with x minus"},{"Start":"03:01.850 ","End":"03:05.555","Text":"2 when we substitute 2 in x minus 2, and we get 0."},{"Start":"03:05.555 ","End":"03:07.490","Text":"But remember, we\u0027re not done here."},{"Start":"03:07.490 ","End":"03:11.045","Text":"0 is not the answer to the problem, that was the side exercise."},{"Start":"03:11.045 ","End":"03:14.360","Text":"We go back her, and this e to the asterisk,"},{"Start":"03:14.360 ","End":"03:16.400","Text":"the asterisk was the 0."},{"Start":"03:16.400 ","End":"03:19.745","Text":"We put 0 in here and the answer is e to the 0,"},{"Start":"03:19.745 ","End":"03:21.245","Text":"which is also equal to 1."},{"Start":"03:21.245 ","End":"03:24.390","Text":"1 is the answer, and now we\u0027re done."}],"ID":8473},{"Watched":false,"Name":"Exercise 4","Duration":"3m 48s","ChapterTopicVideoID":8303,"CourseChapterTopicPlaylistID":52081,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8303.jpeg","UploadDate":"2017-01-25T02:24:48.4800000","DurationForVideoObject":"PT3M48S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.185","Text":"Here we have to compute the limit as x goes to infinity of this expression."},{"Start":"00:04.185 ","End":"00:06.000","Text":"Let\u0027s see what kind of a limit we\u0027re dealing with."},{"Start":"00:06.000 ","End":"00:08.710","Text":"First thing we do is substitute and see what\u0027s going on."},{"Start":"00:08.710 ","End":"00:10.455","Text":"If x goes to infinity,"},{"Start":"00:10.455 ","End":"00:12.345","Text":"this exponent goes to infinity,"},{"Start":"00:12.345 ","End":"00:15.020","Text":"and this quotient actually goes to 1."},{"Start":"00:15.020 ","End":"00:17.684","Text":"When we have something like 1 to the power of infinity,"},{"Start":"00:17.684 ","End":"00:19.220","Text":"something to the power of something,"},{"Start":"00:19.220 ","End":"00:21.510","Text":"then we use this formula and put"},{"Start":"00:21.510 ","End":"00:25.680","Text":"any exponent something to the power of something in terms of e to the power of."},{"Start":"00:25.680 ","End":"00:29.865","Text":"If we do this here, where 1 is A and our infinity is B,"},{"Start":"00:29.865 ","End":"00:31.605","Text":"then we get the following:"},{"Start":"00:31.605 ","End":"00:32.985","Text":"e to the power of."},{"Start":"00:32.985 ","End":"00:35.655","Text":"Instead of this thing, we get e^x squared,"},{"Start":"00:35.655 ","End":"00:37.515","Text":"natural log of this thing."},{"Start":"00:37.515 ","End":"00:39.405","Text":"Now, there is a trick here too,"},{"Start":"00:39.405 ","End":"00:41.120","Text":"do a side exercise,"},{"Start":"00:41.120 ","End":"00:43.175","Text":"pull the answer to this 1 asterisk,"},{"Start":"00:43.175 ","End":"00:48.005","Text":"which is the limit without the e to figure out the limit of x squared"},{"Start":"00:48.005 ","End":"00:53.810","Text":"times natural log of x squared plus 1 over x squared minus 1."},{"Start":"00:53.810 ","End":"00:55.475","Text":"When we\u0027ve got the answer to this,"},{"Start":"00:55.475 ","End":"00:58.055","Text":"then our answer will be e to the power of this."},{"Start":"00:58.055 ","End":"01:02.010","Text":"We want e to the power of the asterisk will be our final answer."},{"Start":"01:02.010 ","End":"01:03.570","Text":"This is what we want."},{"Start":"01:03.570 ","End":"01:06.180","Text":"Let\u0027s go to that side exercise."},{"Start":"01:06.180 ","End":"01:09.110","Text":"The side exercise, the asterisk is this."},{"Start":"01:09.110 ","End":"01:11.960","Text":"If we examine this 1 by the substitution,"},{"Start":"01:11.960 ","End":"01:14.840","Text":"then we get x squared goes to infinity."},{"Start":"01:14.840 ","End":"01:17.390","Text":"This thing actually goes to 1."},{"Start":"01:17.390 ","End":"01:21.260","Text":"So we have the limit of the form infinity times 0."},{"Start":"01:21.260 ","End":"01:23.750","Text":"It\u0027s very close to L\u0027Hopital because if we put"},{"Start":"01:23.750 ","End":"01:26.390","Text":"1 of these on the denominator by inverting it,"},{"Start":"01:26.390 ","End":"01:29.045","Text":"will get 0/0 or infinity over infinity."},{"Start":"01:29.045 ","End":"01:31.100","Text":"Now it\u0027s much better to put the x squared on"},{"Start":"01:31.100 ","End":"01:33.515","Text":"the denominator because after we differentiate,"},{"Start":"01:33.515 ","End":"01:36.835","Text":"natural log would disappear if we leave it on the numerator."},{"Start":"01:36.835 ","End":"01:40.190","Text":"Let\u0027s write it in the form of putting the x squared on the bottom and you"},{"Start":"01:40.190 ","End":"01:43.835","Text":"already must know the trick of putting something on the bottom, but inverting it."},{"Start":"01:43.835 ","End":"01:46.205","Text":"That\u0027s just an algebraic equality."},{"Start":"01:46.205 ","End":"01:49.910","Text":"At this point, we finally get 0/0 because"},{"Start":"01:49.910 ","End":"01:53.930","Text":"we computed that this was 1 and natural log of 1 is 0,"},{"Start":"01:53.930 ","End":"01:56.810","Text":"and 0 over and 1 over infinity is 0."},{"Start":"01:56.810 ","End":"01:58.250","Text":"So we have a 0/0,"},{"Start":"01:58.250 ","End":"02:00.680","Text":"we could use L\u0027Hopital continuing here."},{"Start":"02:00.680 ","End":"02:03.800","Text":"Now, as I said, this is in case of 0/0."},{"Start":"02:03.800 ","End":"02:08.030","Text":"Classic L\u0027Hopital, which is what we do is replace this limit with a different limit,"},{"Start":"02:08.030 ","End":"02:10.760","Text":"write L\u0027Hopital 0/0 and we just"},{"Start":"02:10.760 ","End":"02:14.495","Text":"differentiate both top and bottom separately and get a new limit."},{"Start":"02:14.495 ","End":"02:15.680","Text":"For the natural log,"},{"Start":"02:15.680 ","End":"02:18.110","Text":"it\u0027s 1 over this thing times the internal"},{"Start":"02:18.110 ","End":"02:22.205","Text":"derivative and differentiate the bottom minus 2 over x cubed."},{"Start":"02:22.205 ","End":"02:26.450","Text":"After that, just actually do the differentiation here and we use the quotient rule,"},{"Start":"02:26.450 ","End":"02:29.495","Text":"derivative of the top times the bottom, etc."},{"Start":"02:29.495 ","End":"02:31.460","Text":"You\u0027re familiar with the quotient rule."},{"Start":"02:31.460 ","End":"02:33.530","Text":"I\u0027m going to start over here simplifying"},{"Start":"02:33.530 ","End":"02:37.140","Text":"this numerator because I can see 2x cubed canceling with 2x cubed,"},{"Start":"02:37.140 ","End":"02:42.425","Text":"and now we\u0027re left with minus 2x minus 2x is minus 4x for this whole numerator."},{"Start":"02:42.425 ","End":"02:43.940","Text":"Then when I bring this,"},{"Start":"02:43.940 ","End":"02:46.040","Text":"I can take this from the bottom to the to,"},{"Start":"02:46.040 ","End":"02:47.360","Text":"so if I get rid of it here,"},{"Start":"02:47.360 ","End":"02:51.785","Text":"I can put it here as minus x cubed over 2."},{"Start":"02:51.785 ","End":"02:54.860","Text":"This minus and this minus cancel and we\u0027re left"},{"Start":"02:54.860 ","End":"02:57.590","Text":"in the numerator between these 2 things together,"},{"Start":"02:57.590 ","End":"03:00.680","Text":"we multiply them, all goes with 2 twice."},{"Start":"03:00.680 ","End":"03:05.385","Text":"So it\u0027s 2x^4, because this whole thing."},{"Start":"03:05.385 ","End":"03:09.575","Text":"Here and here, this cancels with 1 of these and gives that."},{"Start":"03:09.575 ","End":"03:11.420","Text":"When all is said and done,"},{"Start":"03:11.420 ","End":"03:13.910","Text":"all we\u0027re left with is this. Multiplying out,"},{"Start":"03:13.910 ","End":"03:17.270","Text":"we come to this, and if we multiply this,"},{"Start":"03:17.270 ","End":"03:22.060","Text":"we\u0027re going to get 2x^4 over x^4 minus 1."},{"Start":"03:22.060 ","End":"03:23.450","Text":"To compute this limit,"},{"Start":"03:23.450 ","End":"03:27.870","Text":"all we need is the leading coefficients, and here, it\u0027s 2x^4,"},{"Start":"03:27.870 ","End":"03:29.850","Text":"and here, it\u0027s x^4."},{"Start":"03:29.850 ","End":"03:32.975","Text":"This just leaves us with 2 and that\u0027s going to be the limit."},{"Start":"03:32.975 ","End":"03:36.830","Text":"But this is not the final answer because we still have to go back."},{"Start":"03:36.830 ","End":"03:38.600","Text":"We computed the asterisk,"},{"Start":"03:38.600 ","End":"03:39.860","Text":"which was this limit."},{"Start":"03:39.860 ","End":"03:43.910","Text":"Now what we want is e to the power of that asterisk over e to the power"},{"Start":"03:43.910 ","End":"03:49.020","Text":"of the 2 that we found and this is the answer, e squared."}],"ID":8474},{"Watched":false,"Name":"Exercise 5","Duration":"3m 48s","ChapterTopicVideoID":8304,"CourseChapterTopicPlaylistID":52081,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8304.jpeg","UploadDate":"2017-01-25T02:25:49.0470000","DurationForVideoObject":"PT3M48S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"In this exercise, we have to compute the limit as x"},{"Start":"00:02.820 ","End":"00:06.105","Text":"goes to 0 from the right of x^sine x."},{"Start":"00:06.105 ","End":"00:08.640","Text":"First thing we try is just substituting,"},{"Start":"00:08.640 ","End":"00:11.805","Text":"and in this case we would get 0^0,"},{"Start":"00:11.805 ","End":"00:15.090","Text":"which is not defined; indeterminate, sometimes called."},{"Start":"00:15.090 ","End":"00:20.735","Text":"We use the standard technique for exponents and that is to use this formula from algebra,"},{"Start":"00:20.735 ","End":"00:24.470","Text":"which converts an exponent into a base e exponent,"},{"Start":"00:24.470 ","End":"00:26.345","Text":"and the red correspond to the red,"},{"Start":"00:26.345 ","End":"00:29.735","Text":"the green with the green, and if we just apply this formula to this,"},{"Start":"00:29.735 ","End":"00:35.570","Text":"what we get is e to the power of sine x times natural log of x."},{"Start":"00:35.570 ","End":"00:38.870","Text":"Now, the usual technique that we used when we have e to"},{"Start":"00:38.870 ","End":"00:43.010","Text":"the power of something is to first of all compute the limit without the e,"},{"Start":"00:43.010 ","End":"00:46.895","Text":"and what I mean is, when I find out the limit of this expression,"},{"Start":"00:46.895 ","End":"00:49.265","Text":"which I will denote by asterisk,"},{"Start":"00:49.265 ","End":"00:51.320","Text":"and then once I have this side limit,"},{"Start":"00:51.320 ","End":"00:55.145","Text":"I\u0027ll go back here and put e to the power of asterisk,"},{"Start":"00:55.145 ","End":"00:58.060","Text":"whatever it came out to be and say what that is."},{"Start":"00:58.060 ","End":"01:00.440","Text":"I\u0027m leaving this bit blank for the moment because I\u0027m going"},{"Start":"01:00.440 ","End":"01:02.930","Text":"to come back here after I\u0027ve done this side exercise."},{"Start":"01:02.930 ","End":"01:06.590","Text":"Remember what I got was just took the e out and I\u0027m going to compute this."},{"Start":"01:06.590 ","End":"01:08.090","Text":"Now what kind of limit is this?"},{"Start":"01:08.090 ","End":"01:11.795","Text":"If we put x is 0 plus, sine x is just 0,"},{"Start":"01:11.795 ","End":"01:15.785","Text":"but the natural log of 0 plus is minus infinity."},{"Start":"01:15.785 ","End":"01:20.210","Text":"What we have here is a 0 times minus infinity situation."},{"Start":"01:20.210 ","End":"01:24.590","Text":"In this case, what we do is we tried to bring 1 of these into"},{"Start":"01:24.590 ","End":"01:30.005","Text":"the denominator and make it either a 0 over 0 or infinity over infinity case."},{"Start":"01:30.005 ","End":"01:33.920","Text":"In this case, it\u0027s easier to move the sine x into the denominator,"},{"Start":"01:33.920 ","End":"01:38.240","Text":"because when we would like to differentiate logarithms and then they disappear,"},{"Start":"01:38.240 ","End":"01:41.660","Text":"and trigonometrical functions just stay trigonometrical functions."},{"Start":"01:41.660 ","End":"01:44.555","Text":"In that case, what I\u0027m going to do is write it like this."},{"Start":"01:44.555 ","End":"01:47.360","Text":"This is a standard thing of moving from the top to the bottom,"},{"Start":"01:47.360 ","End":"01:48.590","Text":"you always make it reciprocal,"},{"Start":"01:48.590 ","End":"01:53.780","Text":"multiplying by a fraction is like dividing by the inverse fraction and vice versa."},{"Start":"01:53.780 ","End":"01:55.940","Text":"This is what we get here, and at this point"},{"Start":"01:55.940 ","End":"01:59.375","Text":"the minus infinity we kept with here, but the 0,"},{"Start":"01:59.375 ","End":"02:02.750","Text":"once I put it in the bottom became infinity because if x is"},{"Start":"02:02.750 ","End":"02:07.175","Text":"0 plus, sine x is also 0 plus and 1 over 0 plus is infinity."},{"Start":"02:07.175 ","End":"02:09.950","Text":"Now, we\u0027re ready to use L\u0027Hopital."},{"Start":"02:09.950 ","End":"02:12.845","Text":"What we\u0027ve written here is that using"},{"Start":"02:12.845 ","End":"02:16.580","Text":"L\u0027Hopital for the minus infinity over infinity case,"},{"Start":"02:16.580 ","End":"02:19.438","Text":"we get the limit, still x goes to 0 plus."},{"Start":"02:19.438 ","End":"02:21.875","Text":"But now we\u0027ve replaced this limit with a new limit,"},{"Start":"02:21.875 ","End":"02:24.350","Text":"what you get when you differentiate top and bottom,"},{"Start":"02:24.350 ","End":"02:27.740","Text":"this is the fraction line, to the top is natural log of x,"},{"Start":"02:27.740 ","End":"02:30.800","Text":"it becomes 1 over x when derived and 1 over sine x,"},{"Start":"02:30.800 ","End":"02:33.440","Text":"you can either use the quotient rule or you can use the 1"},{"Start":"02:33.440 ","End":"02:36.484","Text":"over x rule 1 over something, derived"},{"Start":"02:36.484 ","End":"02:39.410","Text":"is minus 1 over that something squared times"},{"Start":"02:39.410 ","End":"02:42.660","Text":"the internal derivative of sine which is cosine."},{"Start":"02:42.660 ","End":"02:45.290","Text":"Next, we need to simplify this,"},{"Start":"02:45.290 ","End":"02:46.925","Text":"and if we simplify it,"},{"Start":"02:46.925 ","End":"02:51.470","Text":"we get the sine squared comes up into the numerator and I\u0027ve chosen to write it as"},{"Start":"02:51.470 ","End":"02:54.290","Text":"sine times sine, and the cosine x"},{"Start":"02:54.290 ","End":"02:57.500","Text":"stays in the bottom and this x from the top comes down to the bottom."},{"Start":"02:57.500 ","End":"03:02.090","Text":"I rearrange it, I get minus sine x over x times sine x over cosine x."},{"Start":"03:02.090 ","End":"03:04.739","Text":"Now, each 1 of these has a limit separately."},{"Start":"03:04.739 ","End":"03:06.140","Text":"If the minus stays as"},{"Start":"03:06.140 ","End":"03:10.565","Text":"a minus and sine x over x goes to 1, it\u0027s a famous limit,"},{"Start":"03:10.565 ","End":"03:12.230","Text":"sine x over cosine x,"},{"Start":"03:12.230 ","End":"03:14.915","Text":"if we substitute x equals 0"},{"Start":"03:14.915 ","End":"03:17.915","Text":"will be 0 over 1, so that\u0027s a 0."},{"Start":"03:17.915 ","End":"03:20.030","Text":"Well, basically because we have the 0 here,"},{"Start":"03:20.030 ","End":"03:24.290","Text":"the whole thing comes out to be 0, and minus 1 times 0,"},{"Start":"03:24.290 ","End":"03:27.230","Text":"which is as you expected, it equals 0."},{"Start":"03:27.230 ","End":"03:32.405","Text":"Now, this is not the end of the exercise because we still have to go back."},{"Start":"03:32.405 ","End":"03:33.860","Text":"This was a side exercise,"},{"Start":"03:33.860 ","End":"03:35.855","Text":"this is what we call the asterisk."},{"Start":"03:35.855 ","End":"03:39.050","Text":"We have to go back up and put e to the 0,"},{"Start":"03:39.050 ","End":"03:43.160","Text":"which is the asterisk from the bottom when we computed this. So the answer is e to the 0,"},{"Start":"03:43.160 ","End":"03:45.650","Text":"but e to the 0 happens to equal 1."},{"Start":"03:45.650 ","End":"03:49.830","Text":"This is the answer to the exercise, and we\u0027re done."}],"ID":8475},{"Watched":false,"Name":"Exercise 6","Duration":"5m 48s","ChapterTopicVideoID":1476,"CourseChapterTopicPlaylistID":52081,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1476.jpeg","UploadDate":"2014-10-01T10:07:31.5170000","DurationForVideoObject":"PT5M48S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.200","Text":"Here we have to find the limit as x goes to 0 of this expression."},{"Start":"00:05.200 ","End":"00:08.170","Text":"First thing to try and do is just substitute,"},{"Start":"00:08.170 ","End":"00:09.995","Text":"maybe there\u0027s no problem."},{"Start":"00:09.995 ","End":"00:14.640","Text":"Well, tangent of 0 is 0,"},{"Start":"00:14.640 ","End":"00:16.515","Text":"so that leaves us with 1 here."},{"Start":"00:16.515 ","End":"00:19.545","Text":"1 over 0 is plus or minus infinity,"},{"Start":"00:19.545 ","End":"00:23.260","Text":"so it\u0027s either 1 to the infinity or 1 to the minus infinity,"},{"Start":"00:23.260 ","End":"00:25.555","Text":"but we\u0027ll proceed the same way."},{"Start":"00:25.555 ","End":"00:33.400","Text":"What we do is use this common formula from algebra and apply it to the original."},{"Start":"00:33.400 ","End":"00:35.095","Text":"You can color coordinate it,"},{"Start":"00:35.095 ","End":"00:38.905","Text":"put this in red and the exponent in green."},{"Start":"00:38.905 ","End":"00:41.140","Text":"If we apply this here,"},{"Start":"00:41.140 ","End":"00:43.390","Text":"then we get the following."}],"ID":1453},{"Watched":false,"Name":"Exercise 7","Duration":"6m 17s","ChapterTopicVideoID":1477,"CourseChapterTopicPlaylistID":52081,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1477.jpeg","UploadDate":"2014-10-01T10:07:44.0030000","DurationForVideoObject":"PT6M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.220","Text":"Here, we have to find the limit as x goes to 0 from the right of sine x^tangent x."},{"Start":"00:08.220 ","End":"00:10.920","Text":"The first thing we do is to try and substitute,"},{"Start":"00:10.920 ","End":"00:13.080","Text":"maybe there\u0027s no problem."},{"Start":"00:13.080 ","End":"00:17.169","Text":"So we get sine 0^tangent 0,"},{"Start":"00:17.210 ","End":"00:20.370","Text":"but both those things are 0,"},{"Start":"00:20.370 ","End":"00:27.705","Text":"so we end up by getting 1 of those indeterminate undefined form 0/0."},{"Start":"00:27.705 ","End":"00:31.530","Text":"Whenever this happens, we use a standard trick."},{"Start":"00:31.530 ","End":"00:39.020","Text":"The standard technique is to use this formula and have even had it color-coded that a is"},{"Start":"00:39.020 ","End":"00:46.760","Text":"the sine x in red and b is the tangent x in green and just applying this formula to this,"},{"Start":"00:46.760 ","End":"00:48.784","Text":"we get the following."},{"Start":"00:48.784 ","End":"00:55.070","Text":"Instead of this, now it\u0027s base e. Now when we have a base e limit,"},{"Start":"00:55.070 ","End":"01:01.040","Text":"the usual technique is just essentially to throw out the e for a moment,"},{"Start":"01:01.040 ","End":"01:06.815","Text":"work out the limit as if there wasn\u0027t an e as a separate exercise,"},{"Start":"01:06.815 ","End":"01:08.860","Text":"which I usually call asterisk,"},{"Start":"01:08.860 ","End":"01:11.180","Text":"and once I get the answer to asterisk,"},{"Start":"01:11.180 ","End":"01:13.910","Text":"I put e to the power of asterisk."},{"Start":"01:13.910 ","End":"01:21.630","Text":"So I\u0027ll show you what I mean and say"},{"Start":"01:21.630 ","End":"01:25.460","Text":"we leave a bit of blank here because we\u0027re going to come back"},{"Start":"01:25.460 ","End":"01:30.095","Text":"to it, and asterisk will be just like this,"},{"Start":"01:30.095 ","End":"01:31.535","Text":"except without the e,"},{"Start":"01:31.535 ","End":"01:34.405","Text":"just the exponent here."},{"Start":"01:34.405 ","End":"01:41.395","Text":"So let\u0027s investigate this for a moment and see what it looks like."},{"Start":"01:41.395 ","End":"01:44.220","Text":"If x goes to 0,"},{"Start":"01:44.220 ","End":"01:46.875","Text":"the tangent x goes to 0."},{"Start":"01:46.875 ","End":"01:52.520","Text":"If x goes to 0 plus, sine x also goes to 0 plus,"},{"Start":"01:52.520 ","End":"01:56.300","Text":"and the natural logarithm of 0 plus is minus infinity."},{"Start":"01:56.300 ","End":"02:03.740","Text":"So what we get is a case of 0 times minus infinity, or could be 0 times infinity."},{"Start":"02:03.740 ","End":"02:13.680","Text":"This is very easily convertible to a L\u0027Hopital form, and what we do is put 1 of these 2 on"},{"Start":"02:13.680 ","End":"02:18.800","Text":"the denominator and the 1 I would put on the denominator would be"},{"Start":"02:18.800 ","End":"02:25.820","Text":"the tangent because we really"},{"Start":"02:25.820 ","End":"02:28.940","Text":"like to get rid of the natural logarithm, and if we leave"},{"Start":"02:28.940 ","End":"02:33.800","Text":"the natural logarithm on the numerator and as is the derivative,"},{"Start":"02:33.800 ","End":"02:35.120","Text":"we\u0027ll get rid of the natural log,"},{"Start":"02:35.120 ","End":"02:36.815","Text":"we\u0027ll have 1 over something."},{"Start":"02:36.815 ","End":"02:42.845","Text":"So if I put something in the tangent x in the denominator,"},{"Start":"02:42.845 ","End":"02:49.780","Text":"it becomes 1 over tangent x, and 1 over tangent x is cotangent of x."},{"Start":"02:49.780 ","End":"02:52.250","Text":"Just in case you\u0027re not sure about this,"},{"Start":"02:52.250 ","End":"02:54.950","Text":"it\u0027s all to do with fractions."},{"Start":"02:54.950 ","End":"02:59.800","Text":"If you multiply by a fraction, or rather"},{"Start":"02:59.800 ","End":"03:03.485","Text":"if you divide by a fraction is like multiplying by the inverse fraction,"},{"Start":"03:03.485 ","End":"03:08.100","Text":"and so and vice versa, or if you put something in the denominator."}],"ID":1454},{"Watched":false,"Name":"Exercise 8","Duration":"5m 41s","ChapterTopicVideoID":1478,"CourseChapterTopicPlaylistID":52081,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1478.jpeg","UploadDate":"2014-10-01T10:08:05.5170000","DurationForVideoObject":"PT5M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.555","Text":"Here, we have to compute the limit as x goes to 0 from the right of x to the tangent x."},{"Start":"00:06.555 ","End":"00:10.454","Text":"Let\u0027s see if it\u0027s a problem just substituting."},{"Start":"00:10.454 ","End":"00:13.215","Text":"Put x equals 0 and this is 0,"},{"Start":"00:13.215 ","End":"00:15.525","Text":"tangent to 0 is also 0,"},{"Start":"00:15.525 ","End":"00:20.985","Text":"so we see that we\u0027re in a case of 0 to the power of 0."},{"Start":"00:20.985 ","End":"00:27.660","Text":"The standard thing to do here is to use the following formula"},{"Start":"00:27.660 ","End":"00:34.370","Text":"from algebra which turns any exponent into an e to the power of something,"},{"Start":"00:34.370 ","End":"00:36.630","Text":"in other words, the base e exponent."},{"Start":"00:36.630 ","End":"00:38.475","Text":"To make it easier,"},{"Start":"00:38.475 ","End":"00:42.330","Text":"the base is in red and the exponent is in green,"},{"Start":"00:42.330 ","End":"00:46.205","Text":"and so this is the formula, you\u0027ve seen it before."},{"Start":"00:46.205 ","End":"00:49.190","Text":"If I apply it to here,"},{"Start":"00:49.190 ","End":"00:52.535","Text":"what I get is instead of x to the tangent x,"},{"Start":"00:52.535 ","End":"00:54.410","Text":"e to the power of tangent of x,"},{"Start":"00:54.410 ","End":"00:56.940","Text":"natural log of x."},{"Start":"00:58.100 ","End":"01:00.990","Text":"We have an e to the power of limit,"},{"Start":"01:00.990 ","End":"01:04.010","Text":"and there\u0027s also a standard technique for that,"},{"Start":"01:04.010 ","End":"01:08.405","Text":"which is basically to compute the limit of just the exponent,"},{"Start":"01:08.405 ","End":"01:11.590","Text":"and usually, I call that the asterisk,"},{"Start":"01:11.590 ","End":"01:15.500","Text":"meaning, the asterisk is the limit of without e"},{"Start":"01:15.500 ","End":"01:18.116","Text":"just the tangent x natural log of x."},{"Start":"01:18.116 ","End":"01:20.540","Text":"I\u0027m leaving this blank deliberately,"},{"Start":"01:20.540 ","End":"01:22.500","Text":"we\u0027re going to compute this limit,"},{"Start":"01:22.500 ","End":"01:25.460","Text":"and when we find the numerical answer for this,"},{"Start":"01:25.460 ","End":"01:30.760","Text":"then to go back here and say e to the power of asterisk and see what that equals."},{"Start":"01:30.760 ","End":"01:32.465","Text":"So we\u0027ll be returning here."},{"Start":"01:32.465 ","End":"01:35.200","Text":"Let\u0027s look at this limit now."},{"Start":"01:35.200 ","End":"01:37.300","Text":"What goes to 0 plus,"},{"Start":"01:37.300 ","End":"01:43.910","Text":"now we know the natural log of 0 plus is minus infinity and that tangent to 0 is 0,"},{"Start":"01:43.910 ","End":"01:48.815","Text":"so what we have here is a case of 0 times minus infinity,"},{"Start":"01:48.815 ","End":"01:55.055","Text":"and this is very easily convertible into either a 0 over 0 or infinity over infinity,"},{"Start":"01:55.055 ","End":"01:59.360","Text":"all you have to do is put 1 of these 2 terms into the denominator,"},{"Start":"01:59.360 ","End":"02:01.730","Text":"but as a reciprocal inverted."},{"Start":"02:01.730 ","End":"02:07.385","Text":"Now, it\u0027s usually best to leave the natural log alone because when we derive it,"},{"Start":"02:07.385 ","End":"02:10.190","Text":"it will become just 1 over, it\u0027ll be simple."},{"Start":"02:10.190 ","End":"02:16.455","Text":"The other way if you make much headway that way,"},{"Start":"02:16.455 ","End":"02:19.805","Text":"1 way goes in, 1 way usually doesn\u0027t go."},{"Start":"02:19.805 ","End":"02:22.700","Text":"If we leave the natural log of x on the top,"},{"Start":"02:22.700 ","End":"02:25.250","Text":"what we get is this, and you might say,"},{"Start":"02:25.250 ","End":"02:27.890","Text":"where does this co-tangent come from?"},{"Start":"02:27.890 ","End":"02:30.530","Text":"Well, I said that we have to put it as the reciprocal,"},{"Start":"02:30.530 ","End":"02:32.735","Text":"this is really 1 over tangent x"},{"Start":"02:32.735 ","End":"02:37.250","Text":"and basic trig formula is that 1 over tangent x is the co-tangent,"},{"Start":"02:37.250 ","End":"02:39.540","Text":"it\u0027s probably its definition."},{"Start":"02:42.290 ","End":"02:48.740","Text":"Anyway, if we put in x equals 0 plus the cotangent of 0"},{"Start":"02:48.740 ","End":"02:56.600","Text":"plus is infinity, because if you look at it also with cosine over sine,"},{"Start":"02:56.600 ","End":"03:03.050","Text":"sine of 0 plus is 0 plus and the cosine of 0 plus is 1,"},{"Start":"03:03.050 ","End":"03:07.130","Text":"so it\u0027s 1 over 0 plus that\u0027s infinity at the bottom."},{"Start":"03:07.130 ","End":"03:15.545","Text":"At the top, also a famous limit is natural log of 0 plus is minus infinity."},{"Start":"03:15.545 ","End":"03:20.480","Text":"What I\u0027m saying is we have a minus infinity over infinity here,"},{"Start":"03:20.480 ","End":"03:22.505","Text":"which is classic for L\u0027Hopital,"},{"Start":"03:22.505 ","End":"03:26.930","Text":"so what I\u0027m going to say is this is equal to this by L\u0027Hopital."},{"Start":"03:26.930 ","End":"03:29.660","Text":"It\u0027s a different limit, it\u0027s not some algebraic manipulation."},{"Start":"03:29.660 ","End":"03:30.710","Text":"It\u0027s a different limit,"},{"Start":"03:30.710 ","End":"03:33.770","Text":"this is L\u0027Hopital\u0027s Rule that the answer stays the"},{"Start":"03:33.770 ","End":"03:37.185","Text":"same for minus infinity over infinity,"},{"Start":"03:37.185 ","End":"03:38.540","Text":"that\u0027s what L\u0027Hopital said,"},{"Start":"03:38.540 ","End":"03:43.490","Text":"if we replace top and bottom by their derivatives respectively."},{"Start":"03:43.490 ","End":"03:46.685","Text":"For natural log of x, we get 1 over x,"},{"Start":"03:46.685 ","End":"03:48.890","Text":"the cotangent x either by"},{"Start":"03:48.890 ","End":"03:53.510","Text":"the formula books or you could try it on your own with cosine over sine,"},{"Start":"03:53.510 ","End":"03:55.910","Text":"getting the derivative and remembering that"},{"Start":"03:55.910 ","End":"03:58.700","Text":"cosine squared plus sine squared is 1, either way,"},{"Start":"03:58.700 ","End":"04:01.580","Text":"I\u0027m not going to get into that and just write the answer,"},{"Start":"04:01.580 ","End":"04:06.070","Text":"which is minus 1 over sine squared x."},{"Start":"04:06.100 ","End":"04:14.205","Text":"From here, what we can do is a little bit of algebra."},{"Start":"04:14.205 ","End":"04:22.190","Text":"Let me say it another way,"},{"Start":"04:22.190 ","End":"04:23.810","Text":"when you divide by a fraction,"},{"Start":"04:23.810 ","End":"04:26.740","Text":"you multiply by the inverse fraction."},{"Start":"04:26.740 ","End":"04:30.695","Text":"That means we can take the minus sine squared and put it on the top"},{"Start":"04:30.695 ","End":"04:32.795","Text":"and this is what we have."},{"Start":"04:32.795 ","End":"04:39.570","Text":"Now, again, if you substitute x equals 0 plus, here we have 0,"},{"Start":"04:39.570 ","End":"04:41.360","Text":"sine of 0 is also 0,"},{"Start":"04:41.360 ","End":"04:43.025","Text":"so we have 0 over 0,"},{"Start":"04:43.025 ","End":"04:48.185","Text":"and now we can use L\u0027Hopital again with a 0 over 0 case"},{"Start":"04:48.185 ","End":"04:51.680","Text":"where we have to differentiate top and bottom."},{"Start":"04:51.680 ","End":"04:54.335","Text":"The derivative of sine squared x,"},{"Start":"04:54.335 ","End":"04:55.850","Text":"well, if it\u0027s just something squared,"},{"Start":"04:55.850 ","End":"05:00.230","Text":"that would be twice that something times the internal derivative,"},{"Start":"05:00.230 ","End":"05:02.565","Text":"and that\u0027s the sine is cosine."},{"Start":"05:02.565 ","End":"05:03.945","Text":"For x, we derive it,"},{"Start":"05:03.945 ","End":"05:06.630","Text":"differentiate it, we just get 1."},{"Start":"05:06.630 ","End":"05:09.465","Text":"Now we can plug in x equals 0,"},{"Start":"05:09.465 ","End":"05:12.380","Text":"and because sine of 0 is 0,"},{"Start":"05:12.380 ","End":"05:14.510","Text":"this whole thing comes out 0."},{"Start":"05:14.510 ","End":"05:17.540","Text":"But you have to remember that this is not the final answer."},{"Start":"05:17.540 ","End":"05:19.390","Text":"Remember there was a blank left up there,"},{"Start":"05:19.390 ","End":"05:23.170","Text":"so we have to scroll back up and go to this,"},{"Start":"05:23.170 ","End":"05:25.730","Text":"and this is the blank place I\u0027m talking about."},{"Start":"05:25.730 ","End":"05:28.670","Text":"Now that we\u0027ve done the asterisk by dropping out the e,"},{"Start":"05:28.670 ","End":"05:30.290","Text":"we can put the e back in,"},{"Start":"05:30.290 ","End":"05:34.040","Text":"and e to the power of the asterisk is e to the power of 0,"},{"Start":"05:34.040 ","End":"05:35.500","Text":"the 0 is from below."},{"Start":"05:35.500 ","End":"05:41.310","Text":"The answer to that is 1, and this is the answer to the exercise and we\u0027re done here."}],"ID":1455},{"Watched":false,"Name":"Exercise 9","Duration":"6m 26s","ChapterTopicVideoID":1480,"CourseChapterTopicPlaylistID":52081,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1480.jpeg","UploadDate":"2014-10-01T10:08:55.9100000","DurationForVideoObject":"PT6M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.810","Text":"In this exercise, we have to find out the limit as x goes to 0 of this expression."},{"Start":"00:06.810 ","End":"00:12.570","Text":"Now, if we substitute x equals 0,"},{"Start":"00:12.570 ","End":"00:17.790","Text":"we see that this part is 0 and this part is 1 and the"},{"Start":"00:17.790 ","End":"00:24.630","Text":"cotangent of 0, it\u0027s infinity;"},{"Start":"00:24.630 ","End":"00:29.009","Text":"it\u0027s cosine x over sine x, which is 1 over 0,"},{"Start":"00:29.009 ","End":"00:31.395","Text":"which is plus or minus infinity,"},{"Start":"00:31.395 ","End":"00:32.730","Text":"but when we square it,"},{"Start":"00:32.730 ","End":"00:35.650","Text":"it\u0027s definitely plus infinity."},{"Start":"00:35.930 ","End":"00:38.225","Text":"In cases like this,"},{"Start":"00:38.225 ","End":"00:41.900","Text":"there\u0027s a standard technique, and that"},{"Start":"00:41.900 ","End":"00:50.179","Text":"standard technique is to use this formula from algebra,"},{"Start":"00:50.179 ","End":"00:54.110","Text":"which converts every exponent into a base e exponent"},{"Start":"00:54.110 ","End":"00:58.880","Text":"and it\u0027s colored so you can see what I\u0027ve taken as A and what I\u0027ve taken as B."},{"Start":"00:58.880 ","End":"01:00.935","Text":"If we apply this formula,"},{"Start":"01:00.935 ","End":"01:05.315","Text":"we just get this expression instead."},{"Start":"01:05.315 ","End":"01:10.160","Text":"Now, the standard trick for the limit of e to the power of something is"},{"Start":"01:10.160 ","End":"01:15.450","Text":"just to take out the e and just take the exponent."},{"Start":"01:33.290 ","End":"01:35.680","Text":"What we do is we take,"},{"Start":"01:35.680 ","End":"01:37.615","Text":"as I said, the limit without the e,"},{"Start":"01:37.615 ","End":"01:40.415","Text":"and here it is, and call it asterisk."},{"Start":"01:40.415 ","End":"01:42.420","Text":"That\u0027s the side exercise."},{"Start":"01:42.420 ","End":"01:44.820","Text":"We\u0027ll solve this, we\u0027ll get all the way to the end,"},{"Start":"01:44.820 ","End":"01:47.260","Text":"find what this limit is,"},{"Start":"01:47.260 ","End":"01:50.455","Text":"then come all the way back up here and then take"},{"Start":"01:50.455 ","End":"01:53.830","Text":"e to the power of that answer that we got down there,"},{"Start":"01:53.830 ","End":"01:55.630","Text":"that\u0027s how we do these things."},{"Start":"01:55.630 ","End":"02:02.565","Text":"Throw out the e, just look at the limit of the exponent, and what do we see here?"},{"Start":"02:02.565 ","End":"02:05.040","Text":"We try substitute x equals 0,"},{"Start":"02:05.040 ","End":"02:10.505","Text":"cotangent of 0, we already said, when it\u0027s squared it\u0027s infinity,"},{"Start":"02:10.505 ","End":"02:17.030","Text":"natural log of 1 plus 0 is natural log of 1 is just 0,"},{"Start":"02:17.030 ","End":"02:25.610","Text":"which means that what we have is an infinity times 0 situation here."},{"Start":"02:26.720 ","End":"02:30.950","Text":"When we have an infinity times 0 situation,"},{"Start":"02:30.950 ","End":"02:35.345","Text":"we usually take 1 of these into the denominator,"},{"Start":"02:35.345 ","End":"02:39.890","Text":"and usually, we leave the natural logarithm on the top."},{"Start":"02:39.890 ","End":"02:43.950","Text":"In which case, this turns into this."},{"Start":"02:43.950 ","End":"02:46.820","Text":"Remember, when cotangent goes to the denominator,"},{"Start":"02:46.820 ","End":"02:50.000","Text":"it inverts and the inverse of cotangent is tangent."},{"Start":"02:50.000 ","End":"02:51.665","Text":"This is cosine over sine,"},{"Start":"02:51.665 ","End":"02:53.375","Text":"and this is sine over cosine,"},{"Start":"02:53.375 ","End":"02:57.815","Text":"you move from the top to the bottom provided you do the reciprocal each time."},{"Start":"02:57.815 ","End":"03:02.190","Text":"At this point, we\u0027re no longer in the infinity times 0,"},{"Start":"03:02.190 ","End":"03:04.970","Text":"the infinity went to the bottom and became 0."},{"Start":"03:04.970 ","End":"03:10.500","Text":"We\u0027re now in a 0 over 0 situation, and that\u0027s classic for L\u0027Hopital."},{"Start":"03:10.500 ","End":"03:13.770","Text":"What we get is the following."},{"Start":"03:13.770 ","End":"03:17.585","Text":"By L\u0027Hopital\u0027s rule for 0 over 0,"},{"Start":"03:17.585 ","End":"03:20.600","Text":"we can replace this limit with a different 1 where we"},{"Start":"03:20.600 ","End":"03:24.890","Text":"differentiate the top here and differentiate the bottom here."},{"Start":"03:24.890 ","End":"03:27.350","Text":"Let\u0027s see, I\u0027ll just briefly go over the details."},{"Start":"03:27.350 ","End":"03:29.245","Text":"Derivative of the top,"},{"Start":"03:29.245 ","End":"03:32.370","Text":"natural log is 1 over that thing,"},{"Start":"03:32.370 ","End":"03:36.050","Text":"but the internal derivative of 1 plus x squared is 2x."},{"Start":"03:36.050 ","End":"03:40.610","Text":"Tangent squared x, something squared is twice that something, but again,"},{"Start":"03:40.610 ","End":"03:43.610","Text":"internal derivative and you might just remember that"},{"Start":"03:43.610 ","End":"03:47.210","Text":"the derivative of tangent is 1 over cosine squared."},{"Start":"03:47.210 ","End":"03:51.935","Text":"Well, if not, you can try it out by yourself or look in a formula book."},{"Start":"03:51.935 ","End":"03:56.060","Text":"Now, we need to do some tidying up."},{"Start":"03:56.060 ","End":"04:02.745","Text":"What we get after the tidying up is the following,"},{"Start":"04:02.745 ","End":"04:08.634","Text":"where basically, what we did was we put the 1 plus x squared on the bottom,"},{"Start":"04:08.634 ","End":"04:11.385","Text":"the 2 canceled with the 2,"},{"Start":"04:11.385 ","End":"04:13.430","Text":"the x stayed here,"},{"Start":"04:13.430 ","End":"04:17.490","Text":"the cosine squared x came up to the top and the tangent state on the bottom,"},{"Start":"04:17.490 ","End":"04:20.275","Text":"and basically, this is what we get."},{"Start":"04:20.275 ","End":"04:26.040","Text":"Now, how to proceed with this?"},{"Start":"04:26.040 ","End":"04:32.400","Text":"Suggestion is, we just done simplification,"},{"Start":"04:32.400 ","End":"04:39.450","Text":"we can\u0027t just put 0 in because we are at the 0 over 0 situation."},{"Start":"04:42.770 ","End":"04:47.060","Text":"Using L\u0027Hopital, again, here won\u0027t do us very much good,"},{"Start":"04:47.060 ","End":"04:50.150","Text":"but what we can do is some algebraic simplification."},{"Start":"04:50.150 ","End":"04:54.380","Text":"For example, we can take x over tangent x separately because this"},{"Start":"04:54.380 ","End":"04:58.665","Text":"is very similar to x over sine x and we know the limit of that,"},{"Start":"04:58.665 ","End":"05:02.165","Text":"and also we can take the trigonometric this thing"},{"Start":"05:02.165 ","End":"05:06.695","Text":"over this thing and just hope for the best."},{"Start":"05:06.695 ","End":"05:16.410","Text":"What do we say? Just split the product up into 2 separate limits:"},{"Start":"05:16.410 ","End":"05:18.425","Text":"the first bit and the second bit."},{"Start":"05:18.425 ","End":"05:23.050","Text":"Now, the first bit actually goes to 1."},{"Start":"05:23.050 ","End":"05:24.650","Text":"The way to see it is, like I said,"},{"Start":"05:24.650 ","End":"05:28.090","Text":"it\u0027s very much like x over sine x."},{"Start":"05:28.090 ","End":"05:31.340","Text":"If you write tangent as sine over cosine,"},{"Start":"05:31.340 ","End":"05:35.570","Text":"this is like x over sine x times cosine x."},{"Start":"05:35.570 ","End":"05:39.170","Text":"Now, x over sine x goes to 1,"},{"Start":"05:39.170 ","End":"05:44.510","Text":"and the cosine of x also goes to 1 when x goes to 0, so this gives us 1."},{"Start":"05:44.510 ","End":"05:48.210","Text":"The other bit, when x goes to 0 also gives us 1"},{"Start":"05:48.210 ","End":"05:52.020","Text":"over 1 because cosine 0 is 1 and 1 plus 0 squared is also 1,"},{"Start":"05:52.020 ","End":"05:53.955","Text":"so everything comes out 1."},{"Start":"05:53.955 ","End":"05:58.385","Text":"Basically, the answer to this is 1 times 1,"},{"Start":"05:58.385 ","End":"06:01.210","Text":"and that happens to equal 1."},{"Start":"06:01.210 ","End":"06:03.765","Text":"At this point, don\u0027t think we\u0027re done,"},{"Start":"06:03.765 ","End":"06:07.655","Text":"you may have forgotten that we have a debt upstairs there."},{"Start":"06:07.655 ","End":"06:09.890","Text":"We have to complete the exercise,"},{"Start":"06:09.890 ","End":"06:11.900","Text":"we just computed the asterisk;"},{"Start":"06:11.900 ","End":"06:13.910","Text":"that\u0027s without the e to the power of."},{"Start":"06:13.910 ","End":"06:19.850","Text":"Now, we have to take that 1 from below and stick it in here as e^1,"},{"Start":"06:19.850 ","End":"06:22.745","Text":"which personally I would have written as just e"},{"Start":"06:22.745 ","End":"06:26.820","Text":"without throwing out the 1 there, but we\u0027re done."}],"ID":1457},{"Watched":false,"Name":"Exercise 10","Duration":"7m 34s","ChapterTopicVideoID":8305,"CourseChapterTopicPlaylistID":52081,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8305.jpeg","UploadDate":"2017-01-25T02:27:56.2300000","DurationForVideoObject":"PT7M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.145","Text":"In this exercise, we have to compute the limit as x goes to 0 of this whole expression."},{"Start":"00:05.145 ","End":"00:10.035","Text":"First thing we do is try and substitute x equals 0 or let x tend to 0."},{"Start":"00:10.035 ","End":"00:12.585","Text":"Now tangent x over x goes to 1."},{"Start":"00:12.585 ","End":"00:13.740","Text":"We\u0027ve done this before,"},{"Start":"00:13.740 ","End":"00:15.030","Text":"but if you don\u0027t recall,"},{"Start":"00:15.030 ","End":"00:18.690","Text":"this is like sine x over cosine x times x."},{"Start":"00:18.690 ","End":"00:20.280","Text":"Now the cosine x goes to 1,"},{"Start":"00:20.280 ","End":"00:22.710","Text":"so really it\u0027s just like the limit of sine x over x,"},{"Start":"00:22.710 ","End":"00:25.050","Text":"which does go to 1 when x goes to 0."},{"Start":"00:25.050 ","End":"00:26.160","Text":"As for the numerator,"},{"Start":"00:26.160 ","End":"00:29.205","Text":"1 over 0 squared is just infinity."},{"Start":"00:29.205 ","End":"00:31.530","Text":"The 0 squared makes it 0 plus,"},{"Start":"00:31.530 ","End":"00:33.749","Text":"and 1 over 0 plus is plus infinity."},{"Start":"00:33.749 ","End":"00:37.005","Text":"What we have is a case of 1 to the power of infinity."},{"Start":"00:37.005 ","End":"00:40.005","Text":"In such cases of something to the power of something,"},{"Start":"00:40.005 ","End":"00:42.000","Text":"we usually apply this rule,"},{"Start":"00:42.000 ","End":"00:45.390","Text":"which convert any exponent into a base e exponent."},{"Start":"00:45.390 ","End":"00:47.600","Text":"We\u0027ve done this before, so I won\u0027t go into great detail,"},{"Start":"00:47.600 ","End":"00:50.540","Text":"but I have made it easier by coloring the base in red,"},{"Start":"00:50.540 ","End":"00:51.920","Text":"the exponent in green,"},{"Start":"00:51.920 ","End":"00:54.455","Text":"and that\u0027s the A to the B and it becomes this,"},{"Start":"00:54.455 ","End":"00:57.525","Text":"which we apply here and becomes this expression."},{"Start":"00:57.525 ","End":"01:01.510","Text":"Now we use the next technique is when we have e to the power of something."},{"Start":"01:01.510 ","End":"01:04.055","Text":"What we do is we take this thing"},{"Start":"01:04.055 ","End":"01:08.540","Text":"separately and let me say before that that the exponent is probably"},{"Start":"01:08.540 ","End":"01:11.330","Text":"better written with the x squared on the bottom because we\u0027re really"},{"Start":"01:11.330 ","End":"01:15.680","Text":"anticipating L\u0027Hopital and either 0 over 0 or infinity over infinity,"},{"Start":"01:15.680 ","End":"01:17.600","Text":"so this is just a rewrite of this."},{"Start":"01:17.600 ","End":"01:19.820","Text":"Now take the limit, without the e,"},{"Start":"01:19.820 ","End":"01:23.315","Text":"take the limit of just the exponent and call it asterisk."},{"Start":"01:23.315 ","End":"01:26.580","Text":"We\u0027ll continue from here to the end and find what this limit is."},{"Start":"01:26.580 ","End":"01:27.710","Text":"Then with that answer,"},{"Start":"01:27.710 ","End":"01:32.060","Text":"come back here and take e to the power of the answer we found below,"},{"Start":"01:32.060 ","End":"01:34.145","Text":"and that will be the whole answer."},{"Start":"01:34.145 ","End":"01:35.840","Text":"Now let\u0027s do the asterisk,"},{"Start":"01:35.840 ","End":"01:37.505","Text":"which is the side exercise,"},{"Start":"01:37.505 ","End":"01:39.170","Text":"and take a look what a limit it is."},{"Start":"01:39.170 ","End":"01:40.700","Text":"Well, we have a fraction. On the bottom,"},{"Start":"01:40.700 ","End":"01:42.665","Text":"we have 0 squared, which is 0."},{"Start":"01:42.665 ","End":"01:46.970","Text":"On the top, tangent x over x is very much like sine x over x."},{"Start":"01:46.970 ","End":"01:50.255","Text":"The only difference is that tangent is sine over cosine."},{"Start":"01:50.255 ","End":"01:53.090","Text":"It\u0027s like having an extra cosine in here, which is 1."},{"Start":"01:53.090 ","End":"01:55.550","Text":"Sine x over x as we know, goes to 1."},{"Start":"01:55.550 ","End":"02:00.035","Text":"What we have here is a natural log of 1 which is also 0."},{"Start":"02:00.035 ","End":"02:05.465","Text":"In other words, we\u0027re now in a 0 over 0 situation and we can apply L\u0027Hopital."},{"Start":"02:05.465 ","End":"02:07.070","Text":"Before we apply L\u0027Hopital,"},{"Start":"02:07.070 ","End":"02:09.590","Text":"in case you forgotten the formula for how to do"},{"Start":"02:09.590 ","End":"02:12.440","Text":"the derivative of natural log of some function of x,"},{"Start":"02:12.440 ","End":"02:16.940","Text":"I\u0027ll just write this formula where this tangent x over x is like the box."},{"Start":"02:16.940 ","End":"02:19.865","Text":"When we derive it, we get 1 over whatever that is,"},{"Start":"02:19.865 ","End":"02:21.769","Text":"but times the internal derivative,"},{"Start":"02:21.769 ","End":"02:23.760","Text":"a derivative of what was in the box."},{"Start":"02:23.760 ","End":"02:26.870","Text":"Now we definitely say that what we\u0027re doing is we\u0027re going"},{"Start":"02:26.870 ","End":"02:30.095","Text":"from this limit to a different limit because of L\u0027Hopital\u0027s rule."},{"Start":"02:30.095 ","End":"02:33.620","Text":"L\u0027hopital\u0027s rule for 0 over 0 says that what we can do is"},{"Start":"02:33.620 ","End":"02:37.490","Text":"take this and differentiate the top separately and the bottom separately,"},{"Start":"02:37.490 ","End":"02:39.380","Text":"and that will give us our answer."},{"Start":"02:39.380 ","End":"02:42.670","Text":"Natural log of something is 1 over that something,"},{"Start":"02:42.670 ","End":"02:44.770","Text":"so we have 1 over tangent x over x."},{"Start":"02:44.770 ","End":"02:47.390","Text":"But now we need the internal derivative,"},{"Start":"02:47.390 ","End":"02:50.180","Text":"which is the derivative of tangent x over x. I\u0027ll just put"},{"Start":"02:50.180 ","End":"02:54.335","Text":"a prime sign here indicating that I\u0027m going to do the actual differentiation later."},{"Start":"02:54.335 ","End":"02:57.215","Text":"The bottom x squared we get for that 2x."},{"Start":"02:57.215 ","End":"02:59.570","Text":"Now we need to do with this prime."},{"Start":"02:59.570 ","End":"03:01.850","Text":"This prime is a quotient,"},{"Start":"03:01.850 ","End":"03:03.560","Text":"so we\u0027re going to use the quotient rule."},{"Start":"03:03.560 ","End":"03:08.915","Text":"See, I\u0027ve left the 2x the same and the 1 over tangent x over x also just as it is."},{"Start":"03:08.915 ","End":"03:10.760","Text":"For this, I use the quotient rule,"},{"Start":"03:10.760 ","End":"03:13.040","Text":"which means the tangent x gives us 1 over"},{"Start":"03:13.040 ","End":"03:15.870","Text":"cosine squared and then we take the other and then minus,"},{"Start":"03:15.870 ","End":"03:17.060","Text":"in the second term,"},{"Start":"03:17.060 ","End":"03:23.090","Text":"we leave the top as it is and differentiate the bottom so we get 1 times tangent x."},{"Start":"03:23.090 ","End":"03:25.220","Text":"Then we always have the denominator squared."},{"Start":"03:25.220 ","End":"03:27.755","Text":"This is what we get at this point."},{"Start":"03:27.755 ","End":"03:30.635","Text":"We have to figure out this limit and we\u0027re going to"},{"Start":"03:30.635 ","End":"03:34.190","Text":"use the help of a formula for tangent x over x."},{"Start":"03:34.190 ","End":"03:39.110","Text":"Like I said, it\u0027s very much like sine x over a tangent x over x is 1."},{"Start":"03:39.110 ","End":"03:42.950","Text":"That gives us that if we simplify things here."},{"Start":"03:42.950 ","End":"03:47.900","Text":"We still have the 1 over cosine squared times x minus 1 tangent x."},{"Start":"03:47.900 ","End":"03:49.865","Text":"We still have the x squared here,"},{"Start":"03:49.865 ","End":"03:55.160","Text":"but we take a tangent x over x and throw it to the bottom, the 2x."},{"Start":"03:55.160 ","End":"04:01.460","Text":"What we get, since we\u0027re taking our x to 0 and since tangent x over x goes to 1,"},{"Start":"04:01.460 ","End":"04:06.770","Text":"I\u0027ve just basically thrown out this bit and we\u0027re left with what was otherwise there."},{"Start":"04:06.770 ","End":"04:11.180","Text":"Then we can simplify things further by putting the x squared on the bottom,"},{"Start":"04:11.180 ","End":"04:14.480","Text":"making it 2x cubed and x goes over here,"},{"Start":"04:14.480 ","End":"04:20.340","Text":"so it\u0027s x over cosine squared x. the tangent x I\u0027ll write as a sine x over cosine x."},{"Start":"04:20.340 ","End":"04:24.139","Text":"After this, what I can do is put this thing over a common denominator,"},{"Start":"04:24.139 ","End":"04:25.595","Text":"which is cosine squared."},{"Start":"04:25.595 ","End":"04:26.960","Text":"The x stays as it is,"},{"Start":"04:26.960 ","End":"04:31.730","Text":"but here we have to put an extra cosine x to compensate for the fact that we did so here,"},{"Start":"04:31.730 ","End":"04:33.680","Text":"and so we keep working at it."},{"Start":"04:33.680 ","End":"04:37.670","Text":"What we\u0027re going to do then is put the cosine squared x down at the bottom."},{"Start":"04:37.670 ","End":"04:42.920","Text":"In this expression, there\u0027s a formula that sine 2x is twice this."},{"Start":"04:42.920 ","End":"04:44.165","Text":"I\u0027m using this formula,"},{"Start":"04:44.165 ","End":"04:48.125","Text":"and if I divide it by 2 by putting a half in front of it,"},{"Start":"04:48.125 ","End":"04:51.020","Text":"then I will just get sine x times cosine x,"},{"Start":"04:51.020 ","End":"04:53.540","Text":"and that gives me the half sine 2x."},{"Start":"04:53.540 ","End":"04:57.545","Text":"What happens if we put x equals 0 here and try to do that?"},{"Start":"04:57.545 ","End":"05:00.680","Text":"Well, at the bottom we have an x cubed and that makes it 0."},{"Start":"05:00.680 ","End":"05:05.120","Text":"The top x is 0 and also sine of twice 0 is 0,"},{"Start":"05:05.120 ","End":"05:07.385","Text":"so we do have 0 over 0."},{"Start":"05:07.385 ","End":"05:12.275","Text":"It is a case for a L\u0027Hopital for 0 over 0 and get a new limit,"},{"Start":"05:12.275 ","End":"05:15.470","Text":"which is obtained by differentiating top and bottom separately."},{"Start":"05:15.470 ","End":"05:17.435","Text":"At the top, the x gives me 1,"},{"Start":"05:17.435 ","End":"05:21.680","Text":"sine 2x gives me cosine 2x times the internal derivative,"},{"Start":"05:21.680 ","End":"05:23.835","Text":"which is 2, which cancels with this 2."},{"Start":"05:23.835 ","End":"05:25.130","Text":"That\u0027s what we\u0027re left at the top."},{"Start":"05:25.130 ","End":"05:26.780","Text":"At the bottom, we have a product."},{"Start":"05:26.780 ","End":"05:29.090","Text":"We derive the first and get 6x"},{"Start":"05:29.090 ","End":"05:32.360","Text":"squared and leave the other as it is and then the other way round;"},{"Start":"05:32.360 ","End":"05:35.585","Text":"we differentiate the cosine squared x,"},{"Start":"05:35.585 ","End":"05:40.160","Text":"which gives us something squared that\u0027s twice something times the internal derivative,"},{"Start":"05:40.160 ","End":"05:43.830","Text":"which is minus sine x and the 2x cubed always was there."},{"Start":"05:43.830 ","End":"05:46.125","Text":"A little bit of simplification now."},{"Start":"05:46.125 ","End":"05:47.255","Text":"From here to here,"},{"Start":"05:47.255 ","End":"05:51.290","Text":"I use the formulas from trigonometry that cosine of"},{"Start":"05:51.290 ","End":"05:55.560","Text":"2x is equal to 1 minus y sine squared x."},{"Start":"05:55.560 ","End":"05:58.550","Text":"If I use this then 2 sine squared x,"},{"Start":"05:58.550 ","End":"06:01.520","Text":"I bring it to the left equals 1 minus cosine 2x,"},{"Start":"06:01.520 ","End":"06:02.900","Text":"which I bring to the right."},{"Start":"06:02.900 ","End":"06:08.000","Text":"Over here, I see that I have 2x squared and both of these terms as a plus here."},{"Start":"06:08.000 ","End":"06:09.845","Text":"I can take 2x squared out of here,"},{"Start":"06:09.845 ","End":"06:13.640","Text":"and that leaves me with just 3 here times cosine squared x."},{"Start":"06:13.640 ","End":"06:16.010","Text":"If I take 2x squared out of here,"},{"Start":"06:16.010 ","End":"06:18.200","Text":"I\u0027ll be left with x instead of 2x cubed,"},{"Start":"06:18.200 ","End":"06:23.790","Text":"so x here, that goes with the minus and the cosine x and the sine x."},{"Start":"06:23.790 ","End":"06:29.400","Text":"Again, I would have got here minus x times 2 cosine x sine x."},{"Start":"06:29.400 ","End":"06:31.565","Text":"This is a formula we already discussed."},{"Start":"06:31.565 ","End":"06:34.235","Text":"This thing here is sine of 2x."},{"Start":"06:34.235 ","End":"06:38.625","Text":"I replaced that by the sine 2x here and this is what got left with."},{"Start":"06:38.625 ","End":"06:42.270","Text":"Next thing we get is split this into 2 bits."},{"Start":"06:42.270 ","End":"06:45.500","Text":"Now because I know the limit of sine x over x,"},{"Start":"06:45.500 ","End":"06:50.135","Text":"which is 1, the 2 cancels with the 2 and we get sine squared x over x squared."},{"Start":"06:50.135 ","End":"06:53.585","Text":"The other bit we\u0027ll be left with is 1 over this mess here."},{"Start":"06:53.585 ","End":"06:56.840","Text":"Continuing further, we now just can substitute,"},{"Start":"06:56.840 ","End":"06:58.715","Text":"except for this limit which is 1,"},{"Start":"06:58.715 ","End":"07:03.170","Text":"because sine x over x goes to 1 so the square of it goes to 1 squared."},{"Start":"07:03.170 ","End":"07:04.910","Text":"This thing, 1 is here."},{"Start":"07:04.910 ","End":"07:07.025","Text":"At the bottom, if I put x is 0,"},{"Start":"07:07.025 ","End":"07:08.630","Text":"this term goes to 0."},{"Start":"07:08.630 ","End":"07:11.405","Text":"Cosine of 0 is 1 and we\u0027re left with just the 3."},{"Start":"07:11.405 ","End":"07:14.015","Text":"Finally, we get an answer here of 1/3."},{"Start":"07:14.015 ","End":"07:17.210","Text":"But that\u0027s not the final answer for this exercise because we have to"},{"Start":"07:17.210 ","End":"07:20.540","Text":"go back up again to the place we left blank."},{"Start":"07:20.540 ","End":"07:24.365","Text":"We computed the limit of this without the e to the power of,"},{"Start":"07:24.365 ","End":"07:28.130","Text":"and now we have to put that e to the power of 1/3."},{"Start":"07:28.130 ","End":"07:30.590","Text":"This is where we put the asterisk back in."},{"Start":"07:30.590 ","End":"07:35.610","Text":"The answer is e to the power of 1/3 or cube root of e. That\u0027s it. We\u0027re done."}],"ID":8476},{"Watched":false,"Name":"Exercise 11","Duration":"3m 21s","ChapterTopicVideoID":8306,"CourseChapterTopicPlaylistID":52081,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8306.jpeg","UploadDate":"2017-03-11T23:15:23.5600000","DurationForVideoObject":"PT3M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.125","Text":"Here, we have to find the limit as x goes to 0 of this expression."},{"Start":"00:04.125 ","End":"00:07.785","Text":"Well, first of all, try substitution x equals 0, so"},{"Start":"00:07.785 ","End":"00:11.805","Text":"cosine of 0 is 1 to the power of 1 over x^4."},{"Start":"00:11.805 ","End":"00:13.290","Text":"Now, if x is 0,"},{"Start":"00:13.290 ","End":"00:15.390","Text":"x^4 is positive 0,"},{"Start":"00:15.390 ","End":"00:18.930","Text":"0 plus, and so this becomes plus infinity."},{"Start":"00:18.930 ","End":"00:23.894","Text":"Altogether, we get 1 of these cases of 1 to the power of infinity with an exponent."},{"Start":"00:23.894 ","End":"00:27.330","Text":"What we usually do is use this formula from algebra which"},{"Start":"00:27.330 ","End":"00:30.915","Text":"converts any exponent into a base e exponent."},{"Start":"00:30.915 ","End":"00:33.630","Text":"If we apply it to this original limit with"},{"Start":"00:33.630 ","End":"00:37.875","Text":"a being cosine of x squared and b being 1 over x^4,"},{"Start":"00:37.875 ","End":"00:40.215","Text":"then we get the following expression."},{"Start":"00:40.215 ","End":"00:43.670","Text":"I\u0027ll just rewrite that with the x^4 on the bottom"},{"Start":"00:43.670 ","End":"00:47.060","Text":"because I\u0027m optimistic and hoping that this will be a L\u0027Hopital,"},{"Start":"00:47.060 ","End":"00:49.985","Text":"either 0 over 0 or infinity over infinity."},{"Start":"00:49.985 ","End":"00:53.150","Text":"Actually, in this case it turns out to be 0 over 0,"},{"Start":"00:53.150 ","End":"00:56.630","Text":"x^4 is obviously 0 and cosine of x squared is"},{"Start":"00:56.630 ","End":"01:00.880","Text":"1 and its logarithm, natural or otherwise is also 0."},{"Start":"01:00.880 ","End":"01:05.180","Text":"We have a 0 over 0 and we\u0027re about to do a L\u0027Hopital,"},{"Start":"01:05.180 ","End":"01:08.720","Text":"except the 0 over 0 is only on the exponent."},{"Start":"01:08.720 ","End":"01:10.610","Text":"If we do the limit of just the exponent,"},{"Start":"01:10.610 ","End":"01:13.670","Text":"I mean without the e, then indeed we have a L\u0027Hopital."},{"Start":"01:13.670 ","End":"01:16.625","Text":"We\u0027ve done this before, we throw out the e temporarily,"},{"Start":"01:16.625 ","End":"01:20.180","Text":"worked on this as a side problem called the answer asterisk,"},{"Start":"01:20.180 ","End":"01:22.295","Text":"and when we get to the asterisk at the end,"},{"Start":"01:22.295 ","End":"01:26.485","Text":"we come back here and say e to the power of that asterisk."},{"Start":"01:26.485 ","End":"01:30.350","Text":"As I said, this is 1 of those 0 over 0 situations."},{"Start":"01:30.350 ","End":"01:33.080","Text":"We\u0027re going to use L\u0027Hopital to change this limit into"},{"Start":"01:33.080 ","End":"01:36.390","Text":"a new limit where we differentiate both top and bottom."},{"Start":"01:36.390 ","End":"01:41.320","Text":"Before that, just in case you forgotten how to differentiate natural log of something,"},{"Start":"01:41.320 ","End":"01:44.930","Text":"this is this formula to remind you that we take 1 over whatever it"},{"Start":"01:44.930 ","End":"01:48.860","Text":"is under the log and then multiply by the internal derivative."},{"Start":"01:48.860 ","End":"01:50.570","Text":"Now let\u0027s do the L\u0027Hopital."},{"Start":"01:50.570 ","End":"01:55.610","Text":"We get L\u0027Hopital for 0 over 0 and we differentiate both top and bottom."},{"Start":"01:55.610 ","End":"01:58.140","Text":"The x^4 just gives us 4x cubed."},{"Start":"01:58.140 ","End":"02:00.515","Text":"For the natural log, with the help of this formula,"},{"Start":"02:00.515 ","End":"02:03.800","Text":"we put 1 over the cosine squared that was here and"},{"Start":"02:03.800 ","End":"02:07.160","Text":"then we have to multiply by the derivative of cosine squared."},{"Start":"02:07.160 ","End":"02:08.375","Text":"If we do that,"},{"Start":"02:08.375 ","End":"02:11.600","Text":"and remember that\u0027s also for cosine, there is a chain rule;"},{"Start":"02:11.600 ","End":"02:15.590","Text":"the cosine of something derived is minus the sine of that something,"},{"Start":"02:15.590 ","End":"02:17.650","Text":"again, times the internal derivative,"},{"Start":"02:17.650 ","End":"02:23.290","Text":"and so deriving the cosine of x squared gives us the derivative of cosine is minus sine."},{"Start":"02:23.290 ","End":"02:27.920","Text":"We have minus sine of x squared and the internal derivative is 2x."},{"Start":"02:27.920 ","End":"02:32.195","Text":"We end up with this and it needs a bit of simplification."},{"Start":"02:32.195 ","End":"02:35.450","Text":"Here\u0027s the minus sine x squared right at the top."},{"Start":"02:35.450 ","End":"02:38.045","Text":"The cosine x squared goes into the bottom,"},{"Start":"02:38.045 ","End":"02:40.535","Text":"2x cancels with 4x cubed,"},{"Start":"02:40.535 ","End":"02:42.800","Text":"just leaving 2x squared."},{"Start":"02:42.800 ","End":"02:45.875","Text":"Here, we can split it into 2 parts,"},{"Start":"02:45.875 ","End":"02:48.935","Text":"take the sine of x squared over x squared,"},{"Start":"02:48.935 ","End":"02:54.390","Text":"which we\u0027ve done before because sine x over x goes to 1, and this is the remainder."},{"Start":"02:54.390 ","End":"02:59.975","Text":"If we now take this limit which is 1 and substitute x equals 0 here,"},{"Start":"02:59.975 ","End":"03:04.100","Text":"we get from here 1, and from here, just by substitution minus 1"},{"Start":"03:04.100 ","End":"03:08.675","Text":"over 2 because the cosine of x squared is just 1 and this is just minus 1/2."},{"Start":"03:08.675 ","End":"03:11.030","Text":"This is not the final answer to the question."},{"Start":"03:11.030 ","End":"03:14.280","Text":"Remember, we still have to go back up just before the asterisk."},{"Start":"03:14.280 ","End":"03:17.000","Text":"We found that this limit is minus 1/2,"},{"Start":"03:17.000 ","End":"03:20.615","Text":"so the answer here will be e to the power of minus 1/2,"},{"Start":"03:20.615 ","End":"03:22.860","Text":"and that\u0027s the answer."}],"ID":8477},{"Watched":false,"Name":"Exercise 12","Duration":"4m 47s","ChapterTopicVideoID":8307,"CourseChapterTopicPlaylistID":52081,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8307.jpeg","UploadDate":"2017-01-25T02:29:21.9270000","DurationForVideoObject":"PT4M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.830","Text":"In this exercise, we have to figure out the limit as x goes to 0 of this expression."},{"Start":"00:04.830 ","End":"00:07.620","Text":"The first thing we usually do is to try"},{"Start":"00:07.620 ","End":"00:10.290","Text":"and see if there\u0027s any problem in just substituting."},{"Start":"00:10.290 ","End":"00:12.420","Text":"If we substitute x equals 0,"},{"Start":"00:12.420 ","End":"00:17.355","Text":"0 to any positive power is still 0 and cosine of 0 is 1."},{"Start":"00:17.355 ","End":"00:18.555","Text":"The base is 1."},{"Start":"00:18.555 ","End":"00:21.860","Text":"The exponent, 0^2n,"},{"Start":"00:21.860 ","End":"00:23.285","Text":"because it\u0027s an even number,"},{"Start":"00:23.285 ","End":"00:27.425","Text":"will always be positive 0 because if x goes to 0,"},{"Start":"00:27.425 ","End":"00:30.110","Text":"either through the positive side or the negative side,"},{"Start":"00:30.110 ","End":"00:32.269","Text":"when it\u0027s squared, it\u0027s always positive."},{"Start":"00:32.269 ","End":"00:36.185","Text":"Basically this is like x goes to 0 from the positive side."},{"Start":"00:36.185 ","End":"00:37.925","Text":"The answer is 0 plus,"},{"Start":"00:37.925 ","End":"00:40.040","Text":"and 1/ 0 plus is plus infinity."},{"Start":"00:40.040 ","End":"00:43.060","Text":"In short, what we get here is 1 to the infinity."},{"Start":"00:43.060 ","End":"00:47.420","Text":"The way to handle this is to use the formula that we use so often."},{"Start":"00:47.420 ","End":"00:50.975","Text":"The colors are matching with the original exercise,"},{"Start":"00:50.975 ","End":"00:52.445","Text":"some base to some power."},{"Start":"00:52.445 ","End":"00:56.030","Text":"We have a way of converting it into a base e exponent,"},{"Start":"00:56.030 ","End":"01:00.890","Text":"substituting it here, what we get is this expression e to the power of that."},{"Start":"01:00.890 ","End":"01:06.020","Text":"Because I\u0027m optimistic and I\u0027m hoping for a case of L\u0027Hopital of 0/0,"},{"Start":"01:06.020 ","End":"01:09.620","Text":"infinity over infinity, I prefer to write this product as a quotient,"},{"Start":"01:09.620 ","End":"01:12.005","Text":"and so I put the x^2n on the bottom."},{"Start":"01:12.005 ","End":"01:16.350","Text":"The standard technique for dealing with the e to the power of something limit,"},{"Start":"01:16.350 ","End":"01:20.300","Text":"and that is we just compute the limit of the exponent without the e,"},{"Start":"01:20.300 ","End":"01:22.535","Text":"and we do it as a side exercise,"},{"Start":"01:22.535 ","End":"01:24.215","Text":"usually call it asterisk."},{"Start":"01:24.215 ","End":"01:26.465","Text":"When we get the answer to the asterisk,"},{"Start":"01:26.465 ","End":"01:27.740","Text":"we come back here,"},{"Start":"01:27.740 ","End":"01:29.630","Text":"and put e to the power of that answer."},{"Start":"01:29.630 ","End":"01:31.985","Text":"We\u0027re going to leave this line for that when we come back,"},{"Start":"01:31.985 ","End":"01:35.375","Text":"and continue to the limit of this thing without the base,"},{"Start":"01:35.375 ","End":"01:36.800","Text":"just the exponent part,"},{"Start":"01:36.800 ","End":"01:38.510","Text":"and that is here."},{"Start":"01:38.510 ","End":"01:40.410","Text":"Let\u0027s look at this limit,"},{"Start":"01:40.410 ","End":"01:42.795","Text":"and see what we can do from this."},{"Start":"01:42.795 ","End":"01:47.465","Text":"We see that the denominator we\u0027ve already computed is 0, and the numerator,"},{"Start":"01:47.465 ","End":"01:51.155","Text":"the cosine of x to the n, we also said that that equals 1,"},{"Start":"01:51.155 ","End":"01:53.630","Text":"and the natural log of 1 is 0."},{"Start":"01:53.630 ","End":"01:56.165","Text":"Actually, we have a 0/0 case here"},{"Start":"01:56.165 ","End":"01:59.495","Text":"as I was hoping, and now we can do a L\u0027Hopital."},{"Start":"01:59.495 ","End":"02:01.160","Text":"Just before the L\u0027Hopital,"},{"Start":"02:01.160 ","End":"02:03.080","Text":"there\u0027s a formula we\u0027re going to use,"},{"Start":"02:03.080 ","End":"02:04.535","Text":"and in case you\u0027ve forgotten it,"},{"Start":"02:04.535 ","End":"02:05.810","Text":"this is the formula."},{"Start":"02:05.810 ","End":"02:07.240","Text":"It follows from the chain rule,"},{"Start":"02:07.240 ","End":"02:10.250","Text":"and the fact that derivative of natural log of x is 1/x,"},{"Start":"02:10.250 ","End":"02:12.635","Text":"we\u0027ve extended it to natural log of something more."},{"Start":"02:12.635 ","End":"02:14.540","Text":"Just 1 over whatever it is times"},{"Start":"02:14.540 ","End":"02:17.525","Text":"the internal derivative where we\u0027ve seen this thing before."},{"Start":"02:17.525 ","End":"02:20.120","Text":"What we get if we apply L\u0027Hopital in"},{"Start":"02:20.120 ","End":"02:24.320","Text":"the 0/0 case over here, and remember what L\u0027Hopital says we do,"},{"Start":"02:24.320 ","End":"02:26.585","Text":"is we replace this with a new limit,"},{"Start":"02:26.585 ","End":"02:29.990","Text":"which is the 1 where we take the derivative of the top separately"},{"Start":"02:29.990 ","End":"02:31.610","Text":"and of the bottom separately."},{"Start":"02:31.610 ","End":"02:35.345","Text":"Here it is, L\u0027Hopital 0/0 case on the top,"},{"Start":"02:35.345 ","End":"02:36.950","Text":"the derivative of the top,"},{"Start":"02:36.950 ","End":"02:38.945","Text":"and the bottom, the derivative of the bottom."},{"Start":"02:38.945 ","End":"02:42.830","Text":"The bottom is just a standard exponent of polynomial."},{"Start":"02:42.830 ","End":"02:44.240","Text":"At the top, we use this formula,"},{"Start":"02:44.240 ","End":"02:46.150","Text":"1 over whatever this cosine,"},{"Start":"02:46.150 ","End":"02:47.740","Text":"and the internal derivative."},{"Start":"02:47.740 ","End":"02:50.180","Text":"At the moment, I just left the internal derivative with"},{"Start":"02:50.180 ","End":"02:52.640","Text":"the prime sign as is, but the next line,"},{"Start":"02:52.640 ","End":"02:56.195","Text":"we\u0027ll open it up using the formula that the cosine of something"},{"Start":"02:56.195 ","End":"03:00.550","Text":"derived is minus the sine of that something times the internal derivative."},{"Start":"03:00.550 ","End":"03:02.420","Text":"That gives us the same as here,"},{"Start":"03:02.420 ","End":"03:05.360","Text":"except that I\u0027ve derived the cosine of x^n,"},{"Start":"03:05.360 ","End":"03:06.740","Text":"to the minus sign,"},{"Start":"03:06.740 ","End":"03:11.885","Text":"and here\u0027s the internal derivative of the x^n. Here we\u0027re going to simplify a bit,"},{"Start":"03:11.885 ","End":"03:14.510","Text":"and just like the rearranged,"},{"Start":"03:14.510 ","End":"03:16.970","Text":"so we put the cosine of x^n"},{"Start":"03:16.970 ","End":"03:19.085","Text":"down to the bottom here,"},{"Start":"03:19.085 ","End":"03:22.030","Text":"and we cancel the n with the n."},{"Start":"03:22.030 ","End":"03:26.315","Text":"Basically, what we\u0027re left with is this expression here."},{"Start":"03:26.315 ","End":"03:29.825","Text":"Once I\u0027m here, I can separate it into 2 bits."},{"Start":"03:29.825 ","End":"03:34.745","Text":"What I\u0027d like to do is since sine x/x is a well-known limit,"},{"Start":"03:34.745 ","End":"03:38.495","Text":"I\u0027m going to break this x^2n minus 1 into 2 bits,"},{"Start":"03:38.495 ","End":"03:41.495","Text":"x^n times x to the n minus 1."},{"Start":"03:41.495 ","End":"03:44.150","Text":"The usual rules of exponents from algebra."},{"Start":"03:44.150 ","End":"03:46.955","Text":"In other words, what I\u0027m saying is we\u0027re going to do this,"},{"Start":"03:46.955 ","End":"03:50.870","Text":"which was basically just splitting this x to the 2n minus 1."},{"Start":"03:50.870 ","End":"03:54.420","Text":"Notice, n plus n minus 1 is 2n minus 1, so it\u0027s okay."},{"Start":"03:54.420 ","End":"03:59.465","Text":"At this point, I can now write it as the limit of the minus is here."},{"Start":"03:59.465 ","End":"04:03.185","Text":"The sine x^n over x^n is here."},{"Start":"04:03.185 ","End":"04:05.075","Text":"This thing canceled out."},{"Start":"04:05.075 ","End":"04:07.640","Text":"What we\u0027re left with is product of 2 limits,"},{"Start":"04:07.640 ","End":"04:10.520","Text":"sine x^n or x^n and the rest of it was minus"},{"Start":"04:10.520 ","End":"04:13.700","Text":"1/2 cosign x^n. The first 1,"},{"Start":"04:13.700 ","End":"04:16.100","Text":"since it\u0027s the power of sine x/x,"},{"Start":"04:16.100 ","End":"04:17.255","Text":"it goes to 1."},{"Start":"04:17.255 ","End":"04:19.670","Text":"This part is no problem in substituting,"},{"Start":"04:19.670 ","End":"04:21.740","Text":"since cosine of 0 is 1,"},{"Start":"04:21.740 ","End":"04:22.850","Text":"which is minus a half."},{"Start":"04:22.850 ","End":"04:25.850","Text":"What we end up with is 1 from here or 1 to the n,"},{"Start":"04:25.850 ","End":"04:27.365","Text":"which is 1 times this."},{"Start":"04:27.365 ","End":"04:29.105","Text":"In other words, it\u0027s minus a half."},{"Start":"04:29.105 ","End":"04:32.390","Text":"Remember, this is not the final answer to the exercise."},{"Start":"04:32.390 ","End":"04:36.845","Text":"We were in the side exercise I called asterisk so our asterisk is minus a half."},{"Start":"04:36.845 ","End":"04:38.585","Text":"This part was the asterisk."},{"Start":"04:38.585 ","End":"04:41.645","Text":"What we need is e to the power of asterisk,"},{"Start":"04:41.645 ","End":"04:43.235","Text":"which was minus a 1/2."},{"Start":"04:43.235 ","End":"04:45.320","Text":"That\u0027s the answer, e^-1/2"},{"Start":"04:45.320 ","End":"04:48.450","Text":"is the answer to the question, and we\u0027re done here"}],"ID":8478},{"Watched":false,"Name":"Exercise 13","Duration":"4m 42s","ChapterTopicVideoID":8308,"CourseChapterTopicPlaylistID":52081,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8308.jpeg","UploadDate":"2017-01-25T02:30:43.3500000","DurationForVideoObject":"PT4M42S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.030","Text":"Here we need to find the limit as x goes to 0 of this expression."},{"Start":"00:04.030 ","End":"00:06.480","Text":"Let\u0027s see what happens if we just try to substitute."},{"Start":"00:06.480 ","End":"00:10.170","Text":"Well, sine x over x does have a limit when x goes to 0,"},{"Start":"00:10.170 ","End":"00:13.610","Text":"it\u0027s not exactly a substitution but we can do the limit and this is known to be 1."},{"Start":"00:13.610 ","End":"00:19.739","Text":"Whereas 1 over x squared when x goes to 0 is plus infinity because when x goes to 0,"},{"Start":"00:19.739 ","End":"00:23.530","Text":"x squared goes to 0 plus and 1 over 0 plus is infinity."},{"Start":"00:23.530 ","End":"00:25.830","Text":"What we have is 1 to the power of infinity when other"},{"Start":"00:25.830 ","End":"00:28.170","Text":"1 of those undefined indeterminate forms."},{"Start":"00:28.170 ","End":"00:30.990","Text":"The standard trick is to use this formula to"},{"Start":"00:30.990 ","End":"00:34.455","Text":"convert this into an e to the power of something expression."},{"Start":"00:34.455 ","End":"00:38.430","Text":"If we just use this formula and the colors are matched to help you out quickly,"},{"Start":"00:38.430 ","End":"00:41.415","Text":"we see we get this by applying the formula."},{"Start":"00:41.415 ","End":"00:42.650","Text":"Once we\u0027re at this point,"},{"Start":"00:42.650 ","End":"00:45.364","Text":"I prefer to write it at the top as a quotient."},{"Start":"00:45.364 ","End":"00:47.690","Text":"I know we\u0027ll need to use L\u0027Hopital on this."},{"Start":"00:47.690 ","End":"00:51.995","Text":"L\u0027Hopital\u0027s good for quotients 0 over 0 or infinity over infinity."},{"Start":"00:51.995 ","End":"00:54.645","Text":"In this case, you\u0027ll see that it will be 0 over 0."},{"Start":"00:54.645 ","End":"00:56.390","Text":"Once we get to this point,"},{"Start":"00:56.390 ","End":"00:58.250","Text":"then there\u0027s a standard trick we use when"},{"Start":"00:58.250 ","End":"01:00.185","Text":"we have the limit of e to the power of something,"},{"Start":"01:00.185 ","End":"01:04.415","Text":"we just first of all compute the limit without the e to the power of just of this."},{"Start":"01:04.415 ","End":"01:07.580","Text":"I mean that we take this limit of just this without"},{"Start":"01:07.580 ","End":"01:11.120","Text":"the e and we call that asterisk as a side exercise."},{"Start":"01:11.120 ","End":"01:13.310","Text":"Now, we\u0027ll continue figuring out what this"},{"Start":"01:13.310 ","End":"01:15.700","Text":"is and when we get the answer we\u0027ll come back here,"},{"Start":"01:15.700 ","End":"01:17.135","Text":"I left a space for this,"},{"Start":"01:17.135 ","End":"01:22.335","Text":"and the answer will be e to the power of whatever that asterisk turns out to be."},{"Start":"01:22.335 ","End":"01:24.470","Text":"Now, I look at this and wonder,"},{"Start":"01:24.470 ","End":"01:25.535","Text":"what do I have here?"},{"Start":"01:25.535 ","End":"01:28.340","Text":"Well, if I put x equals 0, the denominator 0,"},{"Start":"01:28.340 ","End":"01:29.750","Text":"and an x goes to 0,"},{"Start":"01:29.750 ","End":"01:31.535","Text":"sine x over x goes to 1,"},{"Start":"01:31.535 ","End":"01:33.360","Text":"and natural log of 1 is 0."},{"Start":"01:33.360 ","End":"01:36.890","Text":"Here we have a 0 over 0 situation,"},{"Start":"01:36.890 ","End":"01:38.555","Text":"very good for L\u0027Hopital."},{"Start":"01:38.555 ","End":"01:43.100","Text":"Before, L\u0027Hopital just like to remind you a formula that we\u0027re going to use,"},{"Start":"01:43.100 ","End":"01:46.220","Text":"that the derivative of natural log of something is 1 over that something"},{"Start":"01:46.220 ","End":"01:49.340","Text":"but times the internal derivative of that something."},{"Start":"01:49.340 ","End":"01:54.485","Text":"In this case, we get using this formula on this expression 1 over this thing,"},{"Start":"01:54.485 ","End":"01:59.690","Text":"this dividing sign is the big dividing sign over these times the internal derivative."},{"Start":"01:59.690 ","End":"02:00.890","Text":"I won\u0027t do it just yet,"},{"Start":"02:00.890 ","End":"02:04.160","Text":"but I\u0027ll indicate that we have to differentiate this part;"},{"Start":"02:04.160 ","End":"02:06.710","Text":"on the bottom, the derivative is 2x."},{"Start":"02:06.710 ","End":"02:09.200","Text":"Of course, I should have reminded you that L\u0027Hopital\u0027s says"},{"Start":"02:09.200 ","End":"02:11.840","Text":"that when you have 1 of those 0 over 0 situations,"},{"Start":"02:11.840 ","End":"02:14.015","Text":"you replace this limit with a separate limit."},{"Start":"02:14.015 ","End":"02:15.860","Text":"This is not some algebraic manipulation."},{"Start":"02:15.860 ","End":"02:17.825","Text":"This is application of L\u0027Hopital\u0027s Rule."},{"Start":"02:17.825 ","End":"02:20.285","Text":"We do the differentiation of top and bottom separately,"},{"Start":"02:20.285 ","End":"02:23.845","Text":"a different limit, but it\u0027s guaranteed to have the same answer as the original 1."},{"Start":"02:23.845 ","End":"02:25.895","Text":"Now, continuing with this,"},{"Start":"02:25.895 ","End":"02:28.385","Text":"we have to just do the differentiation here."},{"Start":"02:28.385 ","End":"02:31.340","Text":"I\u0027ve used the quotient rule that when something over something,"},{"Start":"02:31.340 ","End":"02:36.005","Text":"so it\u0027s the derivative of the top times the bottom minus the other way around."},{"Start":"02:36.005 ","End":"02:40.840","Text":"The derivative of the bottom times the top over the bottom squared and we get"},{"Start":"02:40.840 ","End":"02:42.905","Text":"this expression which the limit of"},{"Start":"02:42.905 ","End":"02:46.230","Text":"this well-known limit goes to 1 so we can replace it by 1."},{"Start":"02:46.230 ","End":"02:48.600","Text":"That simplifies this thing here,"},{"Start":"02:48.600 ","End":"02:53.900","Text":"so it is a simplification to just throw out this part and we\u0027re left with this part here."},{"Start":"02:53.900 ","End":"02:59.000","Text":"Now, I can do some algebra here and throw the x squared over to the bottom."},{"Start":"02:59.000 ","End":"03:01.865","Text":"If I throw the x squared on the bottom together with the 2x,"},{"Start":"03:01.865 ","End":"03:05.615","Text":"it will give me 2x cubed and I\u0027ll have a single fraction."},{"Start":"03:05.615 ","End":"03:09.740","Text":"At this point, what I can do is we can apply L\u0027Hopital again,"},{"Start":"03:09.740 ","End":"03:13.010","Text":"because if we try to put in x equals 0, the bottom is 0."},{"Start":"03:13.010 ","End":"03:16.280","Text":"Here, this is 0, which doesn\u0027t matter what cosine of 0 is,"},{"Start":"03:16.280 ","End":"03:18.560","Text":"that\u0027s a 0 and sine 0 is also 0,"},{"Start":"03:18.560 ","End":"03:20.810","Text":"so we have a 0 over 0 situation."},{"Start":"03:20.810 ","End":"03:24.815","Text":"Yes, we shall apply L\u0027Hopital\u0027s rule for the 0 over 0 case,"},{"Start":"03:24.815 ","End":"03:27.260","Text":"and replace this limit within a different limit where we"},{"Start":"03:27.260 ","End":"03:29.960","Text":"derive the top separately and the bottom separately."},{"Start":"03:29.960 ","End":"03:32.060","Text":"The bottom is easier, so I\u0027ll go for that first."},{"Start":"03:32.060 ","End":"03:33.635","Text":"That gives us a 6x squared."},{"Start":"03:33.635 ","End":"03:35.315","Text":"On the top, we have 2 terms."},{"Start":"03:35.315 ","End":"03:36.725","Text":"The first 1 is a product,"},{"Start":"03:36.725 ","End":"03:38.090","Text":"and we\u0027ll use the product rule,"},{"Start":"03:38.090 ","End":"03:42.095","Text":"derivative of this times this plus this times the derivative of this, we get this."},{"Start":"03:42.095 ","End":"03:45.410","Text":"For the second term, sine x its derivative is cosine x,"},{"Start":"03:45.410 ","End":"03:47.075","Text":"but there\u0027s still a minus in front of it."},{"Start":"03:47.075 ","End":"03:48.440","Text":"Now, we\u0027re up to here."},{"Start":"03:48.440 ","End":"03:50.480","Text":"Now, if we simplify this,"},{"Start":"03:50.480 ","End":"03:53.795","Text":"notice that cosine x minus cosine x goes."},{"Start":"03:53.795 ","End":"03:57.410","Text":"Also, we can simplify by canceling this x with 1 of"},{"Start":"03:57.410 ","End":"04:01.350","Text":"these X\u0027s that we\u0027re left with minus sine x over 6x."},{"Start":"04:01.350 ","End":"04:05.210","Text":"Again, L\u0027Hopital, derive the top minus cosine of x,"},{"Start":"04:05.210 ","End":"04:06.970","Text":"derive the bottom just 6."},{"Start":"04:06.970 ","End":"04:09.185","Text":"If we substitute x equals 0,"},{"Start":"04:09.185 ","End":"04:10.565","Text":"cosine x being 1,"},{"Start":"04:10.565 ","End":"04:11.885","Text":"we get minus 1/6."},{"Start":"04:11.885 ","End":"04:15.065","Text":"We could have also done it by sine x over x goes to 1."},{"Start":"04:15.065 ","End":"04:17.180","Text":"In any event, we get minus 6 and it\u0027s nice"},{"Start":"04:17.180 ","End":"04:19.340","Text":"to know that 2 different ways give us the same answer."},{"Start":"04:19.340 ","End":"04:21.920","Text":"But this is not the solution to the question"},{"Start":"04:21.920 ","End":"04:25.265","Text":"because remember this is this asterisk aside exercise."},{"Start":"04:25.265 ","End":"04:26.810","Text":"I\u0027m going to go all the way back."},{"Start":"04:26.810 ","End":"04:31.100","Text":"There was e to the something can I just figured out the limit of the exponent,"},{"Start":"04:31.100 ","End":"04:33.250","Text":"and now I have to put the e back in,"},{"Start":"04:33.250 ","End":"04:36.750","Text":"e to the power of asterisk and the asterisk was minus a 1/6."},{"Start":"04:36.750 ","End":"04:39.440","Text":"This part is the final answer to the question,"},{"Start":"04:39.440 ","End":"04:43.110","Text":"e to the minus 1/6, and we\u0027re done."}],"ID":8479}],"Thumbnail":null,"ID":52081},{"Name":"Infinity Minus Infinity","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"∞-∞","Duration":"20m 8s","ChapterTopicVideoID":8441,"CourseChapterTopicPlaylistID":52082,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8441.jpeg","UploadDate":"2019-10-29T12:41:39.1270000","DurationForVideoObject":"PT20M8S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.625","Text":"Here we come to the final and fourth Part 4, theory."},{"Start":"00:05.625 ","End":"00:07.965","Text":"Previously in Lesson 1,"},{"Start":"00:07.965 ","End":"00:11.550","Text":"we had the original L\u0027Hopital quotients,"},{"Start":"00:11.550 ","End":"00:14.535","Text":"0 over 0 and infinity over infinity,"},{"Start":"00:14.535 ","End":"00:19.110","Text":"then in Part 2, we learned how to do 0 times infinity by bringing it to one of"},{"Start":"00:19.110 ","End":"00:24.870","Text":"these forms via turning multiplication into division."},{"Start":"00:24.870 ","End":"00:26.850","Text":"Then in Part 3,"},{"Start":"00:26.850 ","End":"00:30.890","Text":"we used formula for something to the power of"},{"Start":"00:30.890 ","End":"00:36.980","Text":"something to apply it to these 3 undefined indeterminate cases,"},{"Start":"00:36.980 ","End":"00:39.140","Text":"infinity to the 0, 0 to the 0,"},{"Start":"00:39.140 ","End":"00:41.690","Text":"and 1 to the power infinity,"},{"Start":"00:41.690 ","End":"00:42.920","Text":"that was less than 3."},{"Start":"00:42.920 ","End":"00:45.605","Text":"Today in Lesson 4,"},{"Start":"00:45.605 ","End":"00:50.015","Text":"we\u0027ll be learning how to do infinity minus infinity."},{"Start":"00:50.015 ","End":"00:53.270","Text":"Just as before, these were just symbols."},{"Start":"00:53.270 ","End":"00:57.980","Text":"What does infinity minus infinity mean?"},{"Start":"00:57.980 ","End":"00:59.930","Text":"These are not really numbers."},{"Start":"00:59.930 ","End":"01:03.155","Text":"What they mean is we want to find out the limit"},{"Start":"01:03.155 ","End":"01:06.695","Text":"when x goes to something of the difference of 2 functions,"},{"Start":"01:06.695 ","End":"01:09.845","Text":"f of x minus g of x,"},{"Start":"01:09.845 ","End":"01:13.565","Text":"in which f goes to infinity,"},{"Start":"01:13.565 ","End":"01:15.700","Text":"in other words, that the limit of the same thing."},{"Start":"01:15.700 ","End":"01:17.690","Text":"Let\u0027s just say x goes to a,"},{"Start":"01:17.690 ","End":"01:19.290","Text":"each of them goes to infinity,"},{"Start":"01:19.290 ","End":"01:22.670","Text":"that the limit of f as x goes to a is infinity,"},{"Start":"01:22.670 ","End":"01:27.950","Text":"and the limit of g of x as x goes to a is also infinity."},{"Start":"01:27.950 ","End":"01:30.360","Text":"That\u0027s what it really means when I say infinity over infinity."},{"Start":"01:30.360 ","End":"01:31.730","Text":"The difference of 2 functions,"},{"Start":"01:31.730 ","End":"01:37.890","Text":"each of which tends to infinity as x tends to whatever it is."},{"Start":"01:38.510 ","End":"01:46.250","Text":"The usual technique for this infinity minus infinity is to do a common denominator."},{"Start":"01:46.250 ","End":"01:47.615","Text":"Usually, there are fractions,"},{"Start":"01:47.615 ","End":"01:49.009","Text":"and we do a common denominator,"},{"Start":"01:49.009 ","End":"01:51.260","Text":"and then convert it to another form,"},{"Start":"01:51.260 ","End":"01:55.340","Text":"and that\u0027s usually 0 over 0 or maybe infinity over infinity."},{"Start":"01:55.340 ","End":"01:58.880","Text":"But we\u0027ll see this best in the examples."},{"Start":"01:58.880 ","End":"02:03.080","Text":"The example will be the limit as x goes to 1 of 1 over"},{"Start":"02:03.080 ","End":"02:06.920","Text":"natural log of x minus 1 over x minus 1."},{"Start":"02:06.920 ","End":"02:12.035","Text":"Well, let\u0027s check that indeed this is infinity minus infinity."},{"Start":"02:12.035 ","End":"02:14.850","Text":"What we have here, basically,"},{"Start":"02:14.850 ","End":"02:18.585","Text":"if x goes to 1, then natural log of x is 0."},{"Start":"02:18.585 ","End":"02:22.050","Text":"We have, use a color,"},{"Start":"02:22.050 ","End":"02:25.904","Text":"1 over 0 minus,"},{"Start":"02:25.904 ","End":"02:28.145","Text":"and if x goes to 1 here,"},{"Start":"02:28.145 ","End":"02:32.040","Text":"then we also have minus 1 over 0."},{"Start":"02:33.170 ","End":"02:37.604","Text":"Now, 1 over 0 can be plus or minus infinity,"},{"Start":"02:37.604 ","End":"02:38.895","Text":"but in this case,"},{"Start":"02:38.895 ","End":"02:41.530","Text":"if x goes to 1 from the right,"},{"Start":"02:41.530 ","End":"02:44.289","Text":"then here we have 0 plus,"},{"Start":"02:44.289 ","End":"02:47.455","Text":"and here we have a 0 plus, in which case,"},{"Start":"02:47.455 ","End":"02:49.780","Text":"if we have a 0 plus,"},{"Start":"02:49.780 ","End":"02:53.970","Text":"then we get infinity minus infinity."},{"Start":"02:53.970 ","End":"02:56.920","Text":"If we have x going from the other side,"},{"Start":"02:56.920 ","End":"02:59.650","Text":"we will have a 0 minus in both cases."},{"Start":"02:59.650 ","End":"03:01.000","Text":"For the 0 minus,"},{"Start":"03:01.000 ","End":"03:03.265","Text":"we\u0027ll get minus infinity,"},{"Start":"03:03.265 ","End":"03:06.555","Text":"minus minus infinity,"},{"Start":"03:06.555 ","End":"03:10.120","Text":"which is if you just take the brackets out and make this plus infinity,"},{"Start":"03:10.120 ","End":"03:12.040","Text":"these are the same, these are equal."},{"Start":"03:12.040 ","End":"03:16.370","Text":"In either case, we get infinity minus infinity."},{"Start":"03:16.880 ","End":"03:23.350","Text":"I\u0027ll write over here just equals infinity minus infinity,"},{"Start":"03:23.350 ","End":"03:25.670","Text":"and then we can delete this stuff."},{"Start":"03:25.670 ","End":"03:31.105","Text":"Now remember, the usual technique is common denominator."},{"Start":"03:31.105 ","End":"03:35.360","Text":"What we\u0027ll do is write this as"},{"Start":"03:35.400 ","End":"03:42.760","Text":"equals a limit x goes to 1,"},{"Start":"03:42.760 ","End":"03:46.870","Text":"I cross-multiply and put the common denominator,"},{"Start":"03:46.870 ","End":"03:54.460","Text":"so we get x minus 1 minus natural log of x from"},{"Start":"03:54.460 ","End":"04:03.145","Text":"here over the product of these 2 over natural log of x times x minus 1."},{"Start":"04:03.145 ","End":"04:04.865","Text":"Now, what do we have here?"},{"Start":"04:04.865 ","End":"04:09.869","Text":"When x is 1, then this is 0 on the denominator,"},{"Start":"04:09.869 ","End":"04:11.760","Text":"so the whole denominator is 0."},{"Start":"04:11.760 ","End":"04:14.880","Text":"If x is 1, 1 minus 1 is 0,"},{"Start":"04:14.880 ","End":"04:16.965","Text":"natural log of 1 is 0."},{"Start":"04:16.965 ","End":"04:23.950","Text":"Basically, we can use L\u0027Hopital in the 0 over 0 form."},{"Start":"04:28.520 ","End":"04:31.370","Text":"I\u0027ll leave this for you to do."},{"Start":"04:31.370 ","End":"04:38.270","Text":"All we wanted to show was how to get from infinity minus infinity to 0 over 0,"},{"Start":"04:38.270 ","End":"04:40.595","Text":"where we can really do L\u0027Hopital."},{"Start":"04:40.595 ","End":"04:43.310","Text":"I\u0027ll let you finish this at home."},{"Start":"04:43.310 ","End":"04:47.855","Text":"I\u0027ve already pre-written the second exercise,"},{"Start":"04:47.855 ","End":"04:53.165","Text":"number 2, as the limit as x goes to 0 of this whole stuff."},{"Start":"04:53.165 ","End":"04:57.380","Text":"Once again, we\u0027re going to get infinity minus infinity."},{"Start":"04:57.380 ","End":"05:03.240","Text":"If you look at it, this is equal to 1 over,"},{"Start":"05:03.240 ","End":"05:06.420","Text":"an e^x is 1 minus 1,"},{"Start":"05:06.420 ","End":"05:08.235","Text":"which is 1 over 0."},{"Start":"05:08.235 ","End":"05:10.350","Text":"This also comes out, if you look at it,"},{"Start":"05:10.350 ","End":"05:16.100","Text":"as 1 over 0 minus 1 over 0."},{"Start":"05:16.100 ","End":"05:21.950","Text":"Also, here it turns out that if x goes to 0 from the right,"},{"Start":"05:21.950 ","End":"05:24.005","Text":"then we get a 0 plus here,"},{"Start":"05:24.005 ","End":"05:26.200","Text":"and 0 plus here."},{"Start":"05:26.200 ","End":"05:28.170","Text":"Conversely, on the left,"},{"Start":"05:28.170 ","End":"05:31.180","Text":"we get a 0 minus and a 0 minus."},{"Start":"05:32.510 ","End":"05:35.540","Text":"Just like in the previous exercise,"},{"Start":"05:35.540 ","End":"05:38.315","Text":"we would get infinity less infinity,"},{"Start":"05:38.315 ","End":"05:41.135","Text":"or minus infinity less minus infinity."},{"Start":"05:41.135 ","End":"05:47.310","Text":"But in either case, it comes out to equal infinity minus infinity."},{"Start":"05:47.310 ","End":"05:50.085","Text":"Let\u0027s just get rid of this stuff."},{"Start":"05:50.085 ","End":"05:51.920","Text":"Now that we have this form,"},{"Start":"05:51.920 ","End":"05:55.715","Text":"we\u0027ll take this advice at the usual technique and try this common denominator."},{"Start":"05:55.715 ","End":"05:58.925","Text":"Let\u0027s try a common denominator here."},{"Start":"05:58.925 ","End":"06:00.950","Text":"These are just symbolically,"},{"Start":"06:00.950 ","End":"06:02.510","Text":"I write infinity minus infinity."},{"Start":"06:02.510 ","End":"06:04.279","Text":"The idea is do the common denominator,"},{"Start":"06:04.279 ","End":"06:11.030","Text":"so this equals the limit as x goes to 0."},{"Start":"06:11.030 ","End":"06:13.620","Text":"Just scroll up a bit."},{"Start":"06:18.320 ","End":"06:21.900","Text":"Cross-multiply over the product."},{"Start":"06:21.900 ","End":"06:25.785","Text":"Natural log of x plus 1 here,"},{"Start":"06:25.785 ","End":"06:31.360","Text":"minus this e^x minus 1 here,"},{"Start":"06:32.480 ","End":"06:35.520","Text":"over the product of these 2,"},{"Start":"06:35.520 ","End":"06:44.690","Text":"which is e^x minus 1 times natural log of x plus 1."},{"Start":"06:44.690 ","End":"06:48.785","Text":"Now, if we put x equals 0 here,"},{"Start":"06:48.785 ","End":"06:52.010","Text":"natural log of 1 is 0,"},{"Start":"06:52.010 ","End":"06:56.480","Text":"and e^0 is 1 minus 1 is 0, 0 minus 0."},{"Start":"06:56.480 ","End":"07:00.840","Text":"This basically comes out to be 0."},{"Start":"07:00.850 ","End":"07:10.665","Text":"In the denominator, natural log of x plus 1 is the natural log of 0 plus 1."},{"Start":"07:10.665 ","End":"07:14.565","Text":"Actually 0 times whatever is, anyway."},{"Start":"07:14.565 ","End":"07:16.530","Text":"We also get 0 on the bottom."},{"Start":"07:16.530 ","End":"07:22.700","Text":"Then we can continue with the L\u0027Hopital 0 over 0 form,"},{"Start":"07:22.700 ","End":"07:24.665","Text":"which we already know how to do."},{"Start":"07:24.665 ","End":"07:26.540","Text":"That\u0027s not the point of this part,"},{"Start":"07:26.540 ","End":"07:28.670","Text":"is just to show how we get to something we already know,"},{"Start":"07:28.670 ","End":"07:32.760","Text":"we already covered 0 over 0 in a different section."},{"Start":"07:33.200 ","End":"07:37.380","Text":"Now, I\u0027d like to say a few more words about infinity minus infinity."},{"Start":"07:37.380 ","End":"07:38.720","Text":"For example, in the last example,"},{"Start":"07:38.720 ","End":"07:40.970","Text":"I don\u0027t know what the answer becomes."},{"Start":"07:40.970 ","End":"07:46.115","Text":"But infinity minus infinity could be anything."},{"Start":"07:46.115 ","End":"07:50.285","Text":"Because this could be a larger infinity than this."},{"Start":"07:50.285 ","End":"07:52.910","Text":"I don\u0027t mean that. Let me just write this."},{"Start":"07:52.910 ","End":"07:55.049","Text":"It could be anything."},{"Start":"07:57.160 ","End":"08:00.620","Text":"It means that the first expression that we had,"},{"Start":"08:00.620 ","End":"08:04.945","Text":"remember that we had f of x minus g of x."},{"Start":"08:04.945 ","End":"08:08.165","Text":"This thing was f of x minus g of x,"},{"Start":"08:08.165 ","End":"08:10.189","Text":"where both of these tend to infinity,"},{"Start":"08:10.189 ","End":"08:14.330","Text":"but f of x could go to infinity faster or slower than g of x,"},{"Start":"08:14.330 ","End":"08:17.030","Text":"and then we\u0027d get different answers."},{"Start":"08:17.030 ","End":"08:20.835","Text":"It\u0027s trivial for me to show you an example, anything."},{"Start":"08:20.835 ","End":"08:22.960","Text":"For example, if you said to me,"},{"Start":"08:22.960 ","End":"08:28.189","Text":"say I want the infinity to minus infinity to be 200,"},{"Start":"08:28.189 ","End":"08:30.440","Text":"let\u0027s say, just an example,"},{"Start":"08:30.440 ","End":"08:37.675","Text":"then I could say that the limit as x goes to anything,"},{"Start":"08:37.675 ","End":"08:42.440","Text":"actually, of x plus 200."},{"Start":"08:42.690 ","End":"08:45.895","Text":"Well, no, I mean x goes to infinity. I\u0027m sorry."},{"Start":"08:45.895 ","End":"08:49.675","Text":"It has to be x plus 200 minus x,"},{"Start":"08:49.675 ","End":"08:51.280","Text":"and say what this is."},{"Start":"08:51.280 ","End":"08:52.360","Text":"Well, I could say yes,"},{"Start":"08:52.360 ","End":"08:53.560","Text":"when x goes to infinity,"},{"Start":"08:53.560 ","End":"08:55.870","Text":"x plus 200 also goes to infinity,"},{"Start":"08:55.870 ","End":"08:57.895","Text":"and then x also goes to infinity,"},{"Start":"08:57.895 ","End":"09:01.165","Text":"so really we have infinity minus infinity."},{"Start":"09:01.165 ","End":"09:05.800","Text":"But in this case, if you actually compute the computation inside this outer bracket,"},{"Start":"09:05.800 ","End":"09:08.800","Text":"you\u0027ll get 200 because the answer is 200,"},{"Start":"09:08.800 ","End":"09:10.660","Text":"and it could be anything you want."},{"Start":"09:10.660 ","End":"09:16.825","Text":"Another example is that you could take, for example,"},{"Start":"09:16.825 ","End":"09:23.175","Text":"x squared minus x in this limit as x goes to infinity,"},{"Start":"09:23.175 ","End":"09:25.875","Text":"and again you\u0027ll get infinity minus infinity."},{"Start":"09:25.875 ","End":"09:30.185","Text":"But this infinity, this goes to infinity much faster."},{"Start":"09:30.185 ","End":"09:32.980","Text":"If you take x equals 10,"},{"Start":"09:32.980 ","End":"09:34.705","Text":"it\u0027s 100 minus 10."},{"Start":"09:34.705 ","End":"09:36.340","Text":"But if you take x as 1,000,"},{"Start":"09:36.340 ","End":"09:38.260","Text":"it\u0027s a million minus 1,000."},{"Start":"09:38.260 ","End":"09:40.105","Text":"The difference keeps getting bigger."},{"Start":"09:40.105 ","End":"09:45.355","Text":"In this case, the limit of infinity minus infinity would be infinity."},{"Start":"09:45.355 ","End":"09:48.310","Text":"I mean each of these is infinity, infinity, infinity,"},{"Start":"09:48.310 ","End":"09:53.545","Text":"infinity, and you can get anything you want from infinity minus infinity."},{"Start":"09:53.545 ","End":"09:56.440","Text":"That was just for sake of emphasis."},{"Start":"09:56.440 ","End":"10:02.355","Text":"I\u0027ll leave this so that we could just, 2nd example."},{"Start":"10:02.355 ","End":"10:05.340","Text":"Now there is 1 last thing I want to mention,"},{"Start":"10:05.340 ","End":"10:14.305","Text":"is that what I said before about the usual thing that we do is a common denominator,"},{"Start":"10:14.305 ","End":"10:16.930","Text":"that\u0027s pretty much what you will encounter."},{"Start":"10:16.930 ","End":"10:20.635","Text":"But I have mapped to say something else for advanced students,"},{"Start":"10:20.635 ","End":"10:26.560","Text":"and those who you are studying advanced at an advanced level calculus,"},{"Start":"10:26.560 ","End":"10:28.910","Text":"and you know who you are,"},{"Start":"10:29.190 ","End":"10:33.999","Text":"I\u0027ll show you what to do in other cases besides common denominator."},{"Start":"10:33.999 ","End":"10:36.655","Text":"Now, the rest of you can stop watching right here,"},{"Start":"10:36.655 ","End":"10:40.100","Text":"only the advanced ones need to continue."},{"Start":"10:40.290 ","End":"10:45.205","Text":"I scroll back up to the place where I wrote the usual technique,"},{"Start":"10:45.205 ","End":"10:49.840","Text":"and a few advanced students will teach another technique."},{"Start":"10:49.840 ","End":"10:52.120","Text":"If the usual technique doesn\u0027t work,"},{"Start":"10:52.120 ","End":"10:55.970","Text":"the next technique I recommend is the following."},{"Start":"10:56.600 ","End":"11:00.630","Text":"Oh, and I forgot to say that this other technique will"},{"Start":"11:00.630 ","End":"11:04.785","Text":"work for the case of a difference of 2 functions."},{"Start":"11:04.785 ","End":"11:07.600","Text":"Something minus something."},{"Start":"11:08.250 ","End":"11:11.410","Text":"Well, here it is, I\u0027ve written it out."},{"Start":"11:11.410 ","End":"11:13.750","Text":"As I mentioned, it\u0027s when you have something minus"},{"Start":"11:13.750 ","End":"11:17.335","Text":"something that this technique doesn\u0027t work, you try the following."},{"Start":"11:17.335 ","End":"11:19.810","Text":"Take the stronger function outside of"},{"Start":"11:19.810 ","End":"11:24.414","Text":"the brackets and apply L\u0027Hopital to what\u0027s inside the brackets."},{"Start":"11:24.414 ","End":"11:26.755","Text":"Now, this may sound like gibberish,"},{"Start":"11:26.755 ","End":"11:30.235","Text":"and I will explain by way of an example what it means."},{"Start":"11:30.235 ","End":"11:35.605","Text":"I\u0027ll just say that we did talk about stronger in the functions that go to infinity."},{"Start":"11:35.605 ","End":"11:38.665","Text":"Some go faster, some go slower."},{"Start":"11:38.665 ","End":"11:44.079","Text":"For example, x squared goes to infinity faster than just x when x goes to infinity,"},{"Start":"11:44.079 ","End":"11:47.440","Text":"or e to the power of x goes quicker than x,"},{"Start":"11:47.440 ","End":"11:49.690","Text":"and we mentioned this."},{"Start":"11:49.690 ","End":"11:54.475","Text":"I\u0027ll use that in this example."},{"Start":"11:54.475 ","End":"11:56.920","Text":"We\u0027ll see it in more detail."},{"Start":"11:56.920 ","End":"12:07.940","Text":"Example will"},{"Start":"12:09.600 ","End":"12:16.135","Text":"be the limit as x goes to"},{"Start":"12:16.135 ","End":"12:24.010","Text":"infinity of x minus the natural log of x."},{"Start":"12:24.010 ","End":"12:27.685","Text":"Now, this is a case of a difference of 2 functions,"},{"Start":"12:27.685 ","End":"12:33.055","Text":"and this goes to infinity a lot faster than this."},{"Start":"12:33.055 ","End":"12:38.590","Text":"For example, suppose it was not natural log but irregular log."},{"Start":"12:38.590 ","End":"12:41.755","Text":"Then the log of 10 is 1,"},{"Start":"12:41.755 ","End":"12:44.260","Text":"the log of 100 is 2,"},{"Start":"12:44.260 ","End":"12:46.900","Text":"the log of 1,000 is 3,"},{"Start":"12:46.900 ","End":"12:49.660","Text":"and you can see that this one goes much faster."},{"Start":"12:49.660 ","End":"12:51.835","Text":"Also with natural log."},{"Start":"12:51.835 ","End":"12:57.520","Text":"For example, say natural log of 10 must be about 3."},{"Start":"12:57.520 ","End":"13:00.025","Text":"Natural log of 100 must be about,"},{"Start":"13:00.025 ","End":"13:03.280","Text":"I don\u0027t know, 6 or even a million."},{"Start":"13:03.280 ","End":"13:06.190","Text":"This one going to be around, I don\u0027t know,"},{"Start":"13:06.190 ","End":"13:11.725","Text":"maybe 10 or a bit more."},{"Start":"13:11.725 ","End":"13:17.290","Text":"Natural log of x goes to infinity much slower than x,"},{"Start":"13:17.290 ","End":"13:19.165","Text":"so this is the stronger one."},{"Start":"13:19.165 ","End":"13:22.690","Text":"This is the one we\u0027re going to take outside the brackets."},{"Start":"13:22.690 ","End":"13:26.200","Text":"We have this as the,"},{"Start":"13:26.200 ","End":"13:28.420","Text":"we can write this as 2 limits,"},{"Start":"13:28.420 ","End":"13:36.055","Text":"or, what I mean to say is very first we can do is take something outside the brackets."},{"Start":"13:36.055 ","End":"13:40.600","Text":"X goes to infinity of x and then take it out,"},{"Start":"13:40.600 ","End":"13:42.010","Text":"and what\u0027s left is 1."},{"Start":"13:42.010 ","End":"13:44.980","Text":"We always get this 1 where the stronger 1 was."},{"Start":"13:44.980 ","End":"13:48.115","Text":"Minus, and now we have to divide this by x,"},{"Start":"13:48.115 ","End":"13:52.660","Text":"natural log of x over x."},{"Start":"13:52.660 ","End":"13:56.425","Text":"Now once you\u0027ve taken this outside the brackets,"},{"Start":"13:56.425 ","End":"13:58.930","Text":"x goes to infinity,"},{"Start":"13:58.930 ","End":"14:02.755","Text":"and we\u0027ll try and figure out just what the brackets go to."},{"Start":"14:02.755 ","End":"14:04.795","Text":"As a separate exercise,"},{"Start":"14:04.795 ","End":"14:07.120","Text":"I\u0027m going to stop here."},{"Start":"14:07.120 ","End":"14:08.965","Text":"Maybe I\u0027ll continue later."},{"Start":"14:08.965 ","End":"14:16.180","Text":"But I\u0027ll do another exercise which will just be this and figure out what is the limit as"},{"Start":"14:16.180 ","End":"14:24.470","Text":"x goes to infinity of 1 minus natural log of x over x."},{"Start":"14:26.070 ","End":"14:33.235","Text":"In this case, the first part will be 1,"},{"Start":"14:33.235 ","End":"14:37.525","Text":"and the second part I can do L\u0027Hopital on,"},{"Start":"14:37.525 ","End":"14:40.075","Text":"so it\u0027s 1 minus,"},{"Start":"14:40.075 ","End":"14:44.170","Text":"and since this is of the case infinity over infinity,"},{"Start":"14:44.170 ","End":"14:47.710","Text":"1 minus the limit,"},{"Start":"14:47.710 ","End":"14:50.530","Text":"I can say here,"},{"Start":"14:50.530 ","End":"14:53.875","Text":"L\u0027Hopital, even though I\u0027ve only done it to the second bit,"},{"Start":"14:53.875 ","End":"14:58.910","Text":"the limit in the case of infinity over infinity,"},{"Start":"14:59.400 ","End":"15:05.560","Text":"so it equals the 1 just stays there limit as x goes to infinity."},{"Start":"15:05.560 ","End":"15:08.785","Text":"Now we take the derivative of top and bottom."},{"Start":"15:08.785 ","End":"15:13.900","Text":"It\u0027s 1 over x over 1, here is 1 over x,"},{"Start":"15:13.900 ","End":"15:16.840","Text":"here is 1, and the limit as x goes to infinity,"},{"Start":"15:16.840 ","End":"15:21.190","Text":"if this is 0, so this whole thing is equal to 1."},{"Start":"15:21.190 ","End":"15:25.120","Text":"Now, if I put this back in here,"},{"Start":"15:25.120 ","End":"15:29.335","Text":"if I know that this goes to infinity and this goes to 1,"},{"Start":"15:29.335 ","End":"15:32.215","Text":"this is infinity times 1,"},{"Start":"15:32.215 ","End":"15:33.400","Text":"and this is one of those forms."},{"Start":"15:33.400 ","End":"15:36.445","Text":"It\u0027s not ambiguous, it just is infinity."},{"Start":"15:36.445 ","End":"15:38.965","Text":"This is equal to infinity,"},{"Start":"15:38.965 ","End":"15:41.530","Text":"and there\u0027s an example of using the technique of"},{"Start":"15:41.530 ","End":"15:44.365","Text":"taking the stronger 1 outside the brackets."},{"Start":"15:44.365 ","End":"15:47.630","Text":"Stronger means goes to infinity faster."},{"Start":"15:50.670 ","End":"15:54.190","Text":"Let\u0027s go on to the next example."},{"Start":"15:54.190 ","End":"15:55.240","Text":"We\u0027ll do another example,"},{"Start":"15:55.240 ","End":"15:57.055","Text":"of course, 1 is not enough."},{"Start":"15:57.055 ","End":"16:01.810","Text":"We\u0027ll do an example number 2,"},{"Start":"16:01.810 ","End":"16:04.584","Text":"and that will be the limit."},{"Start":"16:04.584 ","End":"16:08.110","Text":"Again as x goes to infinity, but instead of this,"},{"Start":"16:08.110 ","End":"16:12.324","Text":"we\u0027ll have e to the power of x minus x."},{"Start":"16:12.324 ","End":"16:14.230","Text":"Once again, we\u0027re stuck."},{"Start":"16:14.230 ","End":"16:16.690","Text":"There\u0027s no obvious way to proceed."},{"Start":"16:16.690 ","End":"16:19.405","Text":"It is an infinity minus infinity case,"},{"Start":"16:19.405 ","End":"16:27.610","Text":"but it\u0027s not something to put a common denominator or anything."},{"Start":"16:27.610 ","End":"16:30.580","Text":"What we\u0027ll do is look at which these 2 is stronger now,"},{"Start":"16:30.580 ","End":"16:32.830","Text":"which do you think x or e to the x?"},{"Start":"16:32.830 ","End":"16:36.295","Text":"Well, it turns out that this is much stronger than this."},{"Start":"16:36.295 ","End":"16:38.050","Text":"If this is 1, this is just e,"},{"Start":"16:38.050 ","End":"16:41.605","Text":"but if this goes to be 3e to the 3,"},{"Start":"16:41.605 ","End":"16:44.845","Text":"still, maybe 20, I don\u0027t know how much it equals."},{"Start":"16:44.845 ","End":"16:46.120","Text":"But if you take, say,"},{"Start":"16:46.120 ","End":"16:49.300","Text":"x is 10, or 100,"},{"Start":"16:49.300 ","End":"16:52.270","Text":"e to the power of a 100 is an enormous number,"},{"Start":"16:52.270 ","End":"16:54.085","Text":"one with many zeros."},{"Start":"16:54.085 ","End":"16:56.125","Text":"This is much stronger than this,"},{"Start":"16:56.125 ","End":"16:58.975","Text":"so this is the one we\u0027ll take outside the brackets,"},{"Start":"16:58.975 ","End":"17:06.970","Text":"and rewrite it as limit as x goes to infinity of e to the x,"},{"Start":"17:06.970 ","End":"17:08.725","Text":"and then what\u0027s left,"},{"Start":"17:08.725 ","End":"17:10.210","Text":"instead of the stronger one,"},{"Start":"17:10.210 ","End":"17:11.905","Text":"we are left with 1,"},{"Start":"17:11.905 ","End":"17:16.250","Text":"and then x over e to the x."},{"Start":"17:17.220 ","End":"17:19.795","Text":"Now, this is equal 2."},{"Start":"17:19.795 ","End":"17:24.475","Text":"Let\u0027s do this 1, the inside the brackets separately."},{"Start":"17:24.475 ","End":"17:26.650","Text":"In fact, we only need to do this bit separately,"},{"Start":"17:26.650 ","End":"17:31.665","Text":"so let\u0027s take this bit separately over here and try to see what that is."},{"Start":"17:31.665 ","End":"17:36.020","Text":"I\u0027ve left and equal here because when I finish doing this one,"},{"Start":"17:36.020 ","End":"17:39.080","Text":"I\u0027ll put it back, substitute back in here."},{"Start":"17:39.080 ","End":"17:45.260","Text":"Limit x goes to infinity. More room here."},{"Start":"17:49.020 ","End":"17:53.990","Text":"Now here we see that we definitely have an infinity over infinity case,"},{"Start":"17:53.990 ","End":"17:55.640","Text":"and so this equals,"},{"Start":"17:55.640 ","End":"17:59.255","Text":"and there\u0027s a L\u0027Hopital here."},{"Start":"17:59.255 ","End":"18:06.060","Text":"A L\u0027Hopital of infinity over infinity."},{"Start":"18:06.060 ","End":"18:09.460","Text":"But just over the bit I put here."},{"Start":"18:09.460 ","End":"18:16.910","Text":"This equals 2, just to differentiate top and bottom,"},{"Start":"18:16.910 ","End":"18:19.350","Text":"and we get 1 at the top,"},{"Start":"18:19.350 ","End":"18:23.050","Text":"and the bottom, e to the x is just e to the x."},{"Start":"18:23.050 ","End":"18:24.460","Text":"Now an x goes to infinity,"},{"Start":"18:24.460 ","End":"18:25.930","Text":"e to the x still goes to infinity,"},{"Start":"18:25.930 ","End":"18:29.285","Text":"but 1 over infinity is equal to 0."},{"Start":"18:29.285 ","End":"18:33.935","Text":"This part over here comes out to be 0."},{"Start":"18:33.935 ","End":"18:37.040","Text":"What I\u0027m left with as x goes to infinity,"},{"Start":"18:37.040 ","End":"18:39.160","Text":"e to the x is infinity,"},{"Start":"18:39.160 ","End":"18:42.585","Text":"and here I have 1 minus 0,"},{"Start":"18:42.585 ","End":"18:45.420","Text":"so e to the infinity times 1,"},{"Start":"18:45.420 ","End":"18:48.470","Text":"so the answer is just infinity."},{"Start":"18:48.470 ","End":"18:51.264","Text":"That\u0027s the 2nd example."},{"Start":"18:51.264 ","End":"18:53.510","Text":"Basically, we\u0027re done here."},{"Start":"18:53.510 ","End":"18:58.495","Text":"But there is a very useful rule of"},{"Start":"18:58.495 ","End":"19:00.800","Text":"which functions are stronger and which functions are"},{"Start":"19:00.800 ","End":"19:04.145","Text":"weaker amongst the common and some of the common functions."},{"Start":"19:04.145 ","End":"19:08.150","Text":"Let me say this, that e to the x is one of the"},{"Start":"19:08.150 ","End":"19:11.960","Text":"strongest that we\u0027ll encounter from the simple basic functions."},{"Start":"19:11.960 ","End":"19:14.435","Text":"This goes as stronger than,"},{"Start":"19:14.435 ","End":"19:16.715","Text":"this symbol for stronger than,"},{"Start":"19:16.715 ","End":"19:18.455","Text":"x to the power of n,"},{"Start":"19:18.455 ","End":"19:21.365","Text":"where n is some whole positive number,"},{"Start":"19:21.365 ","End":"19:23.585","Text":"that\u0027s x squared, x cubed,"},{"Start":"19:23.585 ","End":"19:25.760","Text":"x to the power of 50,"},{"Start":"19:25.760 ","End":"19:34.120","Text":"and this is bigger than the natural log of x,"},{"Start":"19:34.120 ","End":"19:35.679","Text":"which is one of the weakest,"},{"Start":"19:35.679 ","End":"19:38.935","Text":"and this actually works for any positive,"},{"Start":"19:38.935 ","End":"19:44.510","Text":"and it could be a half square root of x."},{"Start":"19:44.510 ","End":"19:47.659","Text":"But this is the order of strength of functions,"},{"Start":"19:47.659 ","End":"19:51.380","Text":"and it\u0027s worth remembering so you don\u0027t have to think each time."},{"Start":"19:51.380 ","End":"19:54.670","Text":"With this, we\u0027re finished not only with part 4,"},{"Start":"19:54.670 ","End":"20:01.480","Text":"but with the entire series if you want to call it that."},{"Start":"20:01.480 ","End":"20:08.630","Text":"Introduction to and the theory behind L\u0027Hopital, so that\u0027s it."}],"ID":8638},{"Watched":false,"Name":"Exercise 1","Duration":"3m 45s","ChapterTopicVideoID":1417,"CourseChapterTopicPlaylistID":52082,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1417.jpeg","UploadDate":"2014-10-01T05:41:31.2170000","DurationForVideoObject":"PT3M45S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.835","Text":"In this exercise, we have to compute the limit as x goes to 0 of 1 over sine x minus 1/x."},{"Start":"00:08.835 ","End":"00:13.860","Text":"First, let\u0027s see if it\u0027s any problem just substituting x equals 0. Well, yes."},{"Start":"00:13.860 ","End":"00:20.320","Text":"There\u0027s 1/0 here and 1/0 here."},{"Start":"00:21.920 ","End":"00:25.980","Text":"Basically, if x goes to 0 from the right,"},{"Start":"00:25.980 ","End":"00:28.310","Text":"if x will be positive,"},{"Start":"00:28.310 ","End":"00:33.350","Text":"then we\u0027ll get 1 over plus 0 minus 1 over plus 0,"},{"Start":"00:33.350 ","End":"00:35.290","Text":"which will be infinity minus infinity."},{"Start":"00:35.290 ","End":"00:36.410","Text":"If we go from the left,"},{"Start":"00:36.410 ","End":"00:42.455","Text":"we\u0027ll have here minus 0 minus and a 0 minus and it\u0027ll be this way."},{"Start":"00:42.455 ","End":"00:47.705","Text":"But in either case it\u0027s not 1 above familiar L\u0027Hopital forms,"},{"Start":"00:47.705 ","End":"00:50.135","Text":"and not 1 that\u0027s easy to convert."},{"Start":"00:50.135 ","End":"00:52.040","Text":"Basically, we know 0/0,"},{"Start":"00:52.040 ","End":"00:55.070","Text":"infinity over infinity, and 0 times infinity,"},{"Start":"00:55.070 ","End":"00:56.825","Text":"and this is not 1 of them."},{"Start":"00:56.825 ","End":"00:59.900","Text":"I suggest we do some algebra on this"},{"Start":"00:59.900 ","End":"01:03.560","Text":"and bring it to a form that we\u0027re more familiar with."},{"Start":"01:03.560 ","End":"01:11.075","Text":"Well, the most obvious thing to do would be to put a common denominator for these."},{"Start":"01:11.075 ","End":"01:14.010","Text":"In which case we would get"},{"Start":"01:14.110 ","End":"01:20.750","Text":"just cross-multiplying x times this sine x times this over the product, we\u0027d get this."},{"Start":"01:20.750 ","End":"01:24.410","Text":"Here, if we put x equals 0,"},{"Start":"01:24.410 ","End":"01:28.865","Text":"we get 0 minus 0 over 0 times 0,"},{"Start":"01:28.865 ","End":"01:31.260","Text":"so we get a 0/0,"},{"Start":"01:31.260 ","End":"01:35.060","Text":"and 0/0 we know how to deal with,"},{"Start":"01:35.060 ","End":"01:38.075","Text":"at least we know how to apply L\u0027Hopital to it."},{"Start":"01:38.075 ","End":"01:43.400","Text":"Applying L\u0027Hopital\u0027s rule, we\u0027re going to get a different limit,"},{"Start":"01:43.400 ","End":"01:48.410","Text":"which is obtained by differentiating numerator and denominator,"},{"Start":"01:48.410 ","End":"01:50.195","Text":"and I write it like this,"},{"Start":"01:50.195 ","End":"01:56.075","Text":"L\u0027Hopital for 0/0, so derivative of the numerator is 1 minus cosine x."},{"Start":"01:56.075 ","End":"01:58.115","Text":"The derivative of the denominator,"},{"Start":"01:58.115 ","End":"02:01.320","Text":"if we use the product rule,"},{"Start":"02:02.080 ","End":"02:07.800","Text":"the derivative of fg is f prime g plus fg prime."},{"Start":"02:07.800 ","End":"02:09.750","Text":"You should know that by now."},{"Start":"02:09.750 ","End":"02:13.700","Text":"Anyway, we get derivative of this times this is that."},{"Start":"02:13.700 ","End":"02:17.300","Text":"This times the derivative of this gives us that."},{"Start":"02:17.300 ","End":"02:20.130","Text":"At this point,"},{"Start":"02:29.480 ","End":"02:33.970","Text":"let\u0027s try substituting just x equals 0,"},{"Start":"02:34.300 ","End":"02:39.565","Text":"and the numerator cosine of 0 is 1, so that\u0027s 0."},{"Start":"02:39.565 ","End":"02:46.955","Text":"Again, this term won\u0027t count because,"},{"Start":"02:46.955 ","End":"02:48.740","Text":"well, it\u0027s 0 anyway,"},{"Start":"02:48.740 ","End":"02:51.485","Text":"because x is 0 and here sine x is 0."},{"Start":"02:51.485 ","End":"02:53.345","Text":"We do get 0/0."},{"Start":"02:53.345 ","End":"02:56.885","Text":"So we\u0027ll have to use L\u0027Hopital again."},{"Start":"02:56.885 ","End":"02:59.485","Text":"That\u0027s not so bad,"},{"Start":"02:59.485 ","End":"03:02.850","Text":"especially when done it for you like this."},{"Start":"03:02.850 ","End":"03:06.735","Text":"Derivative of 1, derivative of minus cosine is sine."},{"Start":"03:06.735 ","End":"03:09.060","Text":"This sine gives us this cosine,"},{"Start":"03:09.060 ","End":"03:12.710","Text":"and this cosine x times x is very similar to what we had here,"},{"Start":"03:12.710 ","End":"03:15.215","Text":"gives us another couple of terms."},{"Start":"03:15.215 ","End":"03:24.635","Text":"At this point, then we can finally actually substitute x equals 0."},{"Start":"03:24.635 ","End":"03:30.485","Text":"Well, this 1 is 0, but this is not 0."},{"Start":"03:30.485 ","End":"03:32.540","Text":"This is 1 plus 1,"},{"Start":"03:32.540 ","End":"03:38.100","Text":"so denominator gives us 2 in any event not 0,"},{"Start":"03:38.100 ","End":"03:39.450","Text":"but the numerator is 0,"},{"Start":"03:39.450 ","End":"03:44.439","Text":"so 0/2 will give us an answer of 0."}],"ID":1425},{"Watched":false,"Name":"Exercise 2","Duration":"4m 24s","ChapterTopicVideoID":1413,"CourseChapterTopicPlaylistID":52082,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1413.jpeg","UploadDate":"2014-10-01T05:39:26.4870000","DurationForVideoObject":"PT4M24S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.250","Text":"Here we have to compute the limit as x goes to 1 of"},{"Start":"00:05.250 ","End":"00:11.100","Text":"1 over natural log of x less 1 over x minus 1."},{"Start":"00:11.100 ","End":"00:14.504","Text":"Now if we try the usual,"},{"Start":"00:14.504 ","End":"00:16.215","Text":"which is to substitute,"},{"Start":"00:16.215 ","End":"00:21.270","Text":"what we\u0027ll get is 1 over natural log of 1,"},{"Start":"00:21.270 ","End":"00:23.860","Text":"which is 1 over 0."},{"Start":"00:25.220 ","End":"00:29.730","Text":"Here we\u0027ll get 1 over 1 minus 1,"},{"Start":"00:29.730 ","End":"00:32.250","Text":"which is also 1 over 0."},{"Start":"00:32.250 ","End":"00:39.270","Text":"Basically depending on whether we go to 1 from the left or from the right,"},{"Start":"00:39.270 ","End":"00:43.760","Text":"we\u0027ll get 1 of these 2 forms,"},{"Start":"00:43.760 ","End":"00:50.960","Text":"infinity minus infinity or minus infinity minus minus infinity which plus infinity."},{"Start":"00:50.960 ","End":"00:56.180","Text":"But neither of these is any good to us for using L\u0027Hopital."},{"Start":"00:56.180 ","End":"01:01.640","Text":"So what we\u0027ll have to do is some algebraic manipulation before."},{"Start":"01:01.640 ","End":"01:05.300","Text":"What we can do is we can do"},{"Start":"01:05.300 ","End":"01:10.550","Text":"the subtraction of the fractions just like we put a common denominator,"},{"Start":"01:10.550 ","End":"01:12.589","Text":"which is the product."},{"Start":"01:12.589 ","End":"01:15.470","Text":"Notice that this is separately."},{"Start":"01:15.470 ","End":"01:20.840","Text":"Perhaps I should even highlight that because it looks like the x goes with there"},{"Start":"01:20.840 ","End":"01:30.395","Text":"and we should really just maybe put an extra brackets here so it doesn\u0027t get confusing."},{"Start":"01:30.395 ","End":"01:38.075","Text":"Very well. Now, if we do it this way,"},{"Start":"01:38.075 ","End":"01:41.180","Text":"then when we substitute x equals 1,"},{"Start":"01:41.180 ","End":"01:47.210","Text":"we get 0 times 0 and here we have 1 minus 1 is 0 minus 0."},{"Start":"01:47.210 ","End":"01:49.800","Text":"Basically, we get 0 over 0."},{"Start":"01:49.800 ","End":"01:57.920","Text":"We can use the L\u0027Hopital\u0027s rule for 0 over 0 and get it as the limit,"},{"Start":"01:57.920 ","End":"02:02.375","Text":"we get a different limit than the original"},{"Start":"02:02.375 ","End":"02:07.175","Text":"by L\u0027Hopital\u0027s rule where we just differentiate top and bottom."},{"Start":"02:07.175 ","End":"02:09.545","Text":"For the top, x gives us 1,"},{"Start":"02:09.545 ","End":"02:11.085","Text":"this gives us nothing,"},{"Start":"02:11.085 ","End":"02:13.470","Text":"this gives us minus 1 over x."},{"Start":"02:13.470 ","End":"02:17.270","Text":"On the bottom, we use the product rule."},{"Start":"02:17.270 ","End":"02:26.434","Text":"Just to remind you quickly again that f times g prime is f prime,"},{"Start":"02:26.434 ","End":"02:30.690","Text":"g minus fg prime."},{"Start":"02:31.030 ","End":"02:36.575","Text":"If we apply it to this,"},{"Start":"02:36.575 ","End":"02:46.400","Text":"then we get 1 over x times this as it is and then the next is this as it is,"},{"Start":"02:46.400 ","End":"02:48.395","Text":"and this derived which is just 1."},{"Start":"02:48.395 ","End":"02:51.895","Text":"We get this expression here."},{"Start":"02:51.895 ","End":"02:58.940","Text":"Now, we can just do"},{"Start":"02:58.940 ","End":"03:05.565","Text":"some simplification."},{"Start":"03:05.565 ","End":"03:12.330","Text":"Just multiply the 1 over x by x and we get 1 and here minus 1 over x."},{"Start":"03:13.250 ","End":"03:17.840","Text":"Once again, if you check what happens when you put x equals 1,"},{"Start":"03:17.840 ","End":"03:20.820","Text":"we again get 0 over 0."},{"Start":"03:21.610 ","End":"03:24.485","Text":"This is easy to see in the numerator."},{"Start":"03:24.485 ","End":"03:31.505","Text":"Here, 1 minus 1 is 0 and the natural log of 1 is also 0,"},{"Start":"03:31.505 ","End":"03:33.965","Text":"1 minus 1 plus 0."},{"Start":"03:33.965 ","End":"03:35.435","Text":"Again, we\u0027ll have to use"},{"Start":"03:35.435 ","End":"03:43.920","Text":"L\u0027Hopital and what we"},{"Start":"03:43.920 ","End":"03:46.805","Text":"get is that the 1 goes to nothing."},{"Start":"03:46.805 ","End":"03:49.400","Text":"This is minus 1 over x squared,"},{"Start":"03:49.400 ","End":"03:50.450","Text":"but with another minus,"},{"Start":"03:50.450 ","End":"03:52.280","Text":"so it\u0027s plus 1 over x squared."},{"Start":"03:52.280 ","End":"03:53.675","Text":"This 1 goes to nothing."},{"Start":"03:53.675 ","End":"03:58.055","Text":"Again, minus 1 over x goes to 1 over x squared and the log goes to this."},{"Start":"03:58.055 ","End":"04:03.180","Text":"This is what we have so far."},{"Start":"04:03.180 ","End":"04:07.635","Text":"At this point, we can actually substitute."},{"Start":"04:07.635 ","End":"04:09.785","Text":"When we put x equals 1,"},{"Start":"04:09.785 ","End":"04:11.570","Text":"then 1 over 1 squared,"},{"Start":"04:11.570 ","End":"04:13.715","Text":"that\u0027s 1, this is 1,"},{"Start":"04:13.715 ","End":"04:16.880","Text":"this is 1, we get 1 over 1 plus 1."},{"Start":"04:16.880 ","End":"04:23.250","Text":"In other words, we get 1/2 and we\u0027re done with this 1."}],"ID":1426},{"Watched":false,"Name":"Exercise 3","Duration":"4m 11s","ChapterTopicVideoID":1414,"CourseChapterTopicPlaylistID":52082,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1414.jpeg","UploadDate":"2014-10-01T05:39:52.6200000","DurationForVideoObject":"PT4M11S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.100","Text":"In this exercise, we have to compute the limit as x approaches 0 from"},{"Start":"00:05.100 ","End":"00:10.575","Text":"the right of this expression involving natural logarithms."},{"Start":"00:10.575 ","End":"00:15.480","Text":"The reason for the 0 plus which means approaching from the right is"},{"Start":"00:15.480 ","End":"00:21.130","Text":"because the natural logarithm is only defined on positive numbers."},{"Start":"00:21.320 ","End":"00:25.380","Text":"The first thing to do is just try substituting."},{"Start":"00:25.380 ","End":"00:31.235","Text":"If x is 0 plus the natural log of 0 plus is minus infinity."},{"Start":"00:31.235 ","End":"00:41.000","Text":"Likewise here, what we would be get less minus infinity which is this,"},{"Start":"00:41.000 ","End":"00:46.055","Text":"and that\u0027s we\u0027ve already seen is no good for us with L\u0027Hopital."},{"Start":"00:46.055 ","End":"00:49.415","Text":"Instead we\u0027ll have to do a bit of algebra first."},{"Start":"00:49.415 ","End":"00:52.325","Text":"Now there is a law of logarithms."},{"Start":"00:52.325 ","End":"00:53.720","Text":"A logarithm of a product,"},{"Start":"00:53.720 ","End":"00:56.310","Text":"logarithm of a quotient."},{"Start":"00:56.560 ","End":"01:00.510","Text":"The logarithm of a quotient is the difference of the logarithms,"},{"Start":"01:00.510 ","End":"01:04.475","Text":"so this is 1 rule we have and it works the other way around."},{"Start":"01:04.475 ","End":"01:07.030","Text":"This is what we have here."},{"Start":"01:07.030 ","End":"01:13.720","Text":"This is true for when the argument of each thing is positive"},{"Start":"01:13.720 ","End":"01:17.060","Text":"for a and b being positive and they are positive because"},{"Start":"01:17.060 ","End":"01:20.810","Text":"x is positive and 3 and 5 are positive,"},{"Start":"01:20.810 ","End":"01:23.885","Text":"so that means that 3x and 5x are positive."},{"Start":"01:23.885 ","End":"01:27.650","Text":"If we now write this using that rule,"},{"Start":"01:27.650 ","End":"01:31.760","Text":"we\u0027ll get that this is the natural log of 3x"},{"Start":"01:31.760 ","End":"01:39.035","Text":"over sine 5x and the same thing holds true here."},{"Start":"01:39.035 ","End":"01:45.965","Text":"Also if 5x is close to 0 and positive and sine of 5x is close to 0 and positive."},{"Start":"01:45.965 ","End":"01:49.125","Text":"In any event, yeah,"},{"Start":"01:49.125 ","End":"01:51.455","Text":"whatever I said still holds."},{"Start":"01:51.455 ","End":"01:55.070","Text":"Now, there is a trick that 1 can do."},{"Start":"01:55.070 ","End":"01:59.820","Text":"If you have the limit of the natural log of something,"},{"Start":"02:01.220 ","End":"02:06.140","Text":"first of all compute the limit of this thing,"},{"Start":"02:06.140 ","End":"02:08.090","Text":"and then take the natural log at the end."},{"Start":"02:08.090 ","End":"02:12.680","Text":"In other words, I could actually write it with a natural logarithm first."},{"Start":"02:12.680 ","End":"02:22.200","Text":"Just a second,"},{"Start":"02:22.730 ","End":"02:29.595","Text":"if we take this limit of this thing as a side exercise,"},{"Start":"02:29.595 ","End":"02:32.735","Text":"and I\u0027ll leave this line blank purposely."},{"Start":"02:32.735 ","End":"02:36.890","Text":"What I\u0027m going to do at the end is after I\u0027ve computed this limit,"},{"Start":"02:36.890 ","End":"02:41.165","Text":"is just to take the natural log of the answer of this."},{"Start":"02:41.165 ","End":"02:45.020","Text":"In other words, what I\u0027m going to do at the end is just take"},{"Start":"02:45.020 ","End":"02:50.750","Text":"the natural log back again and whatever I find in this asterisk,"},{"Start":"02:50.750 ","End":"02:54.720","Text":"that\u0027s what I\u0027m going to write down here."},{"Start":"02:55.810 ","End":"02:59.990","Text":"Let\u0027s continue."},{"Start":"02:59.990 ","End":"03:02.060","Text":"How do I do this limit?"},{"Start":"03:02.060 ","End":"03:06.390","Text":"Well, when I put x equals 0 this is 0,"},{"Start":"03:06.390 ","End":"03:07.815","Text":"sine of 0 is 0,"},{"Start":"03:07.815 ","End":"03:09.090","Text":"3 times 0 is 0."},{"Start":"03:09.090 ","End":"03:12.095","Text":"We have a 0 over 0 limit, in which case,"},{"Start":"03:12.095 ","End":"03:17.130","Text":"now we can use L\u0027Hopital\u0027s rule."},{"Start":"03:18.320 ","End":"03:24.620","Text":"What we\u0027ll get is differentiating the numerator is 3."},{"Start":"03:24.620 ","End":"03:28.865","Text":"Differentiate the denominator instead of sine, we get cosine."},{"Start":"03:28.865 ","End":"03:31.820","Text":"But there\u0027s a chain rule and there\u0027s an inner function,"},{"Start":"03:31.820 ","End":"03:35.280","Text":"so we have to multiply by 5."},{"Start":"03:35.450 ","End":"03:43.715","Text":"Here now we can just straight away substitute cosine of 0 is 1,"},{"Start":"03:43.715 ","End":"03:47.285","Text":"so here we just have 3/5."},{"Start":"03:47.285 ","End":"03:53.190","Text":"Now I\u0027m going to go back here to this point and revisit,"},{"Start":"03:53.190 ","End":"03:56.775","Text":"and the asterisk turned out to be 3/5."},{"Start":"03:56.775 ","End":"04:00.080","Text":"The answer to this problem,"},{"Start":"04:00.080 ","End":"04:02.390","Text":"the solution is over here,"},{"Start":"04:02.390 ","End":"04:04.715","Text":"not the 3/5 down there,"},{"Start":"04:04.715 ","End":"04:11.130","Text":"but this natural log of 3/5 is the answer. We\u0027re done."}],"ID":1427},{"Watched":false,"Name":"Exercise 4","Duration":"6m 6s","ChapterTopicVideoID":1415,"CourseChapterTopicPlaylistID":52082,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1415.jpeg","UploadDate":"2014-10-01T05:40:29.1670000","DurationForVideoObject":"PT6M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.095","Text":"In this exercise, we have to figure out this limit."},{"Start":"00:04.095 ","End":"00:09.180","Text":"Limit is x goes to infinity of the square root of this stuff minus x."},{"Start":"00:09.180 ","End":"00:12.660","Text":"If we substitute x equals infinity,"},{"Start":"00:12.660 ","End":"00:15.150","Text":"we see that we get infinity under"},{"Start":"00:15.150 ","End":"00:18.960","Text":"the square root sign all together infinity minus infinity."},{"Start":"00:18.960 ","End":"00:24.960","Text":"The infinity minus infinity form is undefined,"},{"Start":"00:24.960 ","End":"00:27.660","Text":"so we\u0027re going to have to use some technique."},{"Start":"00:27.660 ","End":"00:33.490","Text":"It turns out that we can use L\u0027Hopital and we will use L\u0027Hopital on this."},{"Start":"00:33.490 ","End":"00:38.925","Text":"But I have to, in all honesty mention that it goes easier with conjugates."},{"Start":"00:38.925 ","End":"00:42.530","Text":"I see a square root and I see 2 expressions instead of the minus,"},{"Start":"00:42.530 ","End":"00:45.950","Text":"I could put the plus and I could solve it very well with conjugates."},{"Start":"00:45.950 ","End":"00:49.100","Text":"But we\u0027re here to practice L\u0027Hopital and so we\u0027ll"},{"Start":"00:49.100 ","End":"00:52.810","Text":"do it with L\u0027Hopital even though the other method is easier."},{"Start":"00:52.810 ","End":"00:55.670","Text":"I\u0027m going to start by doing a little bit of"},{"Start":"00:55.670 ","End":"00:58.640","Text":"algebra to get this into a more convenient form."},{"Start":"00:58.640 ","End":"01:00.125","Text":"Under the square root sign,"},{"Start":"01:00.125 ","End":"01:01.910","Text":"I look for the largest exponent,"},{"Start":"01:01.910 ","End":"01:03.410","Text":"which is the largest power,"},{"Start":"01:03.410 ","End":"01:06.185","Text":"which is x squared, and take it outside the brackets."},{"Start":"01:06.185 ","End":"01:11.180","Text":"If I do that, if I take x squared outside the brackets under the square root,"},{"Start":"01:11.180 ","End":"01:12.980","Text":"I get this expression."},{"Start":"01:12.980 ","End":"01:16.985","Text":"Next thing I\u0027d like to do is to take the x squared and"},{"Start":"01:16.985 ","End":"01:20.980","Text":"bring it out to the brackets after the square root sign."},{"Start":"01:20.980 ","End":"01:24.290","Text":"If I bring the x squared after the square root sign,"},{"Start":"01:24.290 ","End":"01:26.900","Text":"it will become the square root of x squared."},{"Start":"01:26.900 ","End":"01:32.270","Text":"However, when x is positive and it is in our case because it\u0027s going to infinity,"},{"Start":"01:32.270 ","End":"01:36.270","Text":"the square root of x squared is just x."},{"Start":"01:36.310 ","End":"01:39.005","Text":"We take out x squared,"},{"Start":"01:39.005 ","End":"01:43.800","Text":"we\u0027re left with this expression and there\u0027s an x here."},{"Start":"01:43.800 ","End":"01:46.740","Text":"Now look, there\u0027s an x here and an x here."},{"Start":"01:46.740 ","End":"01:55.460","Text":"We can take that x outside of brackets and be left with x times this minus this."},{"Start":"01:55.460 ","End":"02:02.280","Text":"But this is just what is written here."},{"Start":"02:02.590 ","End":"02:05.975","Text":"Now, if I look at this,"},{"Start":"02:05.975 ","End":"02:09.110","Text":"I see that I have infinity times 0,"},{"Start":"02:09.110 ","End":"02:10.985","Text":"x goes to infinity."},{"Start":"02:10.985 ","End":"02:13.880","Text":"As it does, these 2 expressions go to 0."},{"Start":"02:13.880 ","End":"02:16.355","Text":"I get the square root of 1 minus 1."},{"Start":"02:16.355 ","End":"02:20.700","Text":"In brief, I get infinity times 0."},{"Start":"02:20.700 ","End":"02:22.710","Text":"Infinity times 0 is very good,"},{"Start":"02:22.710 ","End":"02:24.560","Text":"it\u0027s very close to L\u0027Hopital."},{"Start":"02:24.560 ","End":"02:29.210","Text":"Because what we do is we just put 1 of these 2 things in the product,"},{"Start":"02:29.210 ","End":"02:32.855","Text":"either the infinity or the 0 into the denominator."},{"Start":"02:32.855 ","End":"02:37.670","Text":"It seems to me that the 1 that should go into the denominator should be the x,"},{"Start":"02:37.670 ","End":"02:39.860","Text":"because it comes out to be much simpler."},{"Start":"02:39.860 ","End":"02:42.260","Text":"Now if you put x into the denominator,"},{"Start":"02:42.260 ","End":"02:48.320","Text":"then what you get is 1 over x because it has to be the reciprocal."},{"Start":"02:48.320 ","End":"02:51.900","Text":"Let me give you an example in arithmetic even."},{"Start":"02:52.460 ","End":"02:55.140","Text":"Instead of this expression,"},{"Start":"02:55.140 ","End":"02:59.570","Text":"and suppose I had 10 times 4, which is 40."},{"Start":"02:59.570 ","End":"03:04.585","Text":"This would be the same as 4 over 1/10."},{"Start":"03:04.585 ","End":"03:06.739","Text":"If you did the other 1 into the denominator,"},{"Start":"03:06.739 ","End":"03:09.815","Text":"it would also be 10 divided by 1/4."},{"Start":"03:09.815 ","End":"03:11.600","Text":"This is the idea of putting something from"},{"Start":"03:11.600 ","End":"03:14.510","Text":"the numerator and the denominator as its reciprocal."},{"Start":"03:14.510 ","End":"03:17.335","Text":"That\u0027s all I did there,"},{"Start":"03:17.335 ","End":"03:20.015","Text":"so I have this thing over 1 over x."},{"Start":"03:20.015 ","End":"03:22.415","Text":"Now if you check this,"},{"Start":"03:22.415 ","End":"03:24.125","Text":"this part we already checked was the,"},{"Start":"03:24.125 ","End":"03:25.535","Text":"was the 0 part,"},{"Start":"03:25.535 ","End":"03:29.840","Text":"but the infinity part has gone into the denominator\u0027s 1 over infinity."},{"Start":"03:29.840 ","End":"03:32.765","Text":"That means that we have 0 over 0 here."},{"Start":"03:32.765 ","End":"03:40.960","Text":"0 over 0 is now excellent for L\u0027Hopital. We have here 0."},{"Start":"03:43.040 ","End":"03:47.150","Text":"Just a second, I just write that down to say that we"},{"Start":"03:47.150 ","End":"03:51.185","Text":"have 0 over 0 and we\u0027re going to use L\u0027Hopital."},{"Start":"03:51.185 ","End":"03:56.610","Text":"But before we do use L\u0027Hopital, we have a square root."},{"Start":"03:56.610 ","End":"04:01.130","Text":"I just wanted to remind you of the derivative of a square root of the formula ,"},{"Start":"04:01.130 ","End":"04:03.410","Text":"follows from the chain rule."},{"Start":"04:03.410 ","End":"04:07.640","Text":"When you take the square root function and you apply it to another function of x,"},{"Start":"04:07.640 ","End":"04:11.180","Text":"the square and it\u0027s the exterior derivative,"},{"Start":"04:11.180 ","End":"04:14.000","Text":"1 over twice square root of that same thing,"},{"Start":"04:14.000 ","End":"04:15.829","Text":"but times the interior derivative,"},{"Start":"04:15.829 ","End":"04:19.435","Text":"which is what this thing is derived prime."},{"Start":"04:19.435 ","End":"04:23.105","Text":"If we do all of this at this expression,"},{"Start":"04:23.105 ","End":"04:25.715","Text":"we\u0027re left with something that looks quite frightening."},{"Start":"04:25.715 ","End":"04:28.385","Text":"But don\u0027t worry, not too bad really."},{"Start":"04:28.385 ","End":"04:31.505","Text":"This square root becomes 1 over twice the square root."},{"Start":"04:31.505 ","End":"04:33.455","Text":"The derivative of the inside,"},{"Start":"04:33.455 ","End":"04:34.940","Text":"you\u0027ll check is this,"},{"Start":"04:34.940 ","End":"04:37.450","Text":"derivative of the bottom is this."},{"Start":"04:37.450 ","End":"04:42.829","Text":"We write equals with a L\u0027Hopital symbol to say that it\u0027s an algebraic equality."},{"Start":"04:42.829 ","End":"04:47.400","Text":"It\u0027s using L\u0027Hopital\u0027s rule for 0 over 0."},{"Start":"04:47.490 ","End":"04:52.100","Text":"Now, let\u0027s see, we\u0027re coming to the end of the page,"},{"Start":"04:52.100 ","End":"04:58.489","Text":"so let\u0027s just turn the page and copy what we had before."},{"Start":"04:58.489 ","End":"05:01.895","Text":"What do we do with this? A bit of algebra."},{"Start":"05:01.895 ","End":"05:04.550","Text":"The denominator of the numerator throw it"},{"Start":"05:04.550 ","End":"05:07.160","Text":"into the denominator and denominator of the denominator,"},{"Start":"05:07.160 ","End":"05:08.420","Text":"throw it into the numerator."},{"Start":"05:08.420 ","End":"05:13.190","Text":"In other words, using the same algebraic tricks of 1 over"},{"Start":"05:13.190 ","End":"05:18.110","Text":"something in the bottom is without the 1 over at the top and vice versa."},{"Start":"05:18.110 ","End":"05:20.255","Text":"Anyway, this is just algebra."},{"Start":"05:20.255 ","End":"05:22.565","Text":"This goes to the bottom, this goes to the top."},{"Start":"05:22.565 ","End":"05:28.055","Text":"Now the first thing to do is multiply this minus x squared by this expression."},{"Start":"05:28.055 ","End":"05:30.500","Text":"The minus will make these 2 minuses go."},{"Start":"05:30.500 ","End":"05:35.480","Text":"The x squared over x squared and 2 x squared over x cubed."},{"Start":"05:35.480 ","End":"05:39.230","Text":"In brief, if you multiply it out using standard algebra,"},{"Start":"05:39.230 ","End":"05:41.260","Text":"we get this expression."},{"Start":"05:41.260 ","End":"05:44.390","Text":"Now this expression is very good because here we can"},{"Start":"05:44.390 ","End":"05:47.540","Text":"finally actually substitute x equals infinity."},{"Start":"05:47.540 ","End":"05:50.360","Text":"When we put x equals infinity,the 1 over x,"},{"Start":"05:50.360 ","End":"05:54.480","Text":"or 1 over x to any power is 0. This is 0."},{"Start":"05:55.550 ","End":"05:58.355","Text":"We are left with this,"},{"Start":"05:58.355 ","End":"06:01.940","Text":"putting this for the zeros and I get rid of the excess zeros."},{"Start":"06:01.940 ","End":"06:04.445","Text":"What you are left with is 1/2,"},{"Start":"06:04.445 ","End":"06:06.630","Text":"and that\u0027s the answer."}],"ID":1428},{"Watched":false,"Name":"Exercise 5","Duration":"6m 41s","ChapterTopicVideoID":1416,"CourseChapterTopicPlaylistID":52082,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1416.jpeg","UploadDate":"2014-10-01T05:41:08.9170000","DurationForVideoObject":"PT6M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.940","Text":"In this exercise, we have to find the limit as"},{"Start":"00:02.940 ","End":"00:05.865","Text":"x goes to minus infinity of this expression."},{"Start":"00:05.865 ","End":"00:10.665","Text":"Now the first thing to do is see what\u0027s the problem with just substitution."},{"Start":"00:10.665 ","End":"00:12.645","Text":"Under the square root sign,"},{"Start":"00:12.645 ","End":"00:17.280","Text":"there\u0027s a slight difficulty because it\u0027s squared,"},{"Start":"00:17.280 ","End":"00:20.750","Text":"it\u0027s plus infinity, and there\u0027s minus infinity plus 1."},{"Start":"00:20.750 ","End":"00:25.040","Text":"But actually what\u0027s under the square root sign goes to infinity, and you can see it."},{"Start":"00:25.040 ","End":"00:32.015","Text":"If you write, do a little bit of algebra and see that x squared plus x plus 1 is"},{"Start":"00:32.015 ","End":"00:40.820","Text":"equal to x squared times 1 plus 1 over x plus 1 over x squared."},{"Start":"00:40.820 ","End":"00:44.015","Text":"Now if x goes to infinity or minus infinity,"},{"Start":"00:44.015 ","End":"00:45.995","Text":"these 2 go to 0,"},{"Start":"00:45.995 ","End":"00:48.364","Text":"and so we end up with x squared,"},{"Start":"00:48.364 ","End":"00:51.995","Text":"which is definitely infinity when x goes to infinity."},{"Start":"00:51.995 ","End":"00:57.560","Text":"What we have here is a case of infinity and also the plus x,"},{"Start":"00:57.560 ","End":"01:01.980","Text":"because x goes to minus infinity is minus infinity."},{"Start":"01:01.980 ","End":"01:06.090","Text":"In other words, we have a case of infinity minus infinity."},{"Start":"01:06.220 ","End":"01:10.850","Text":"Now I\u0027ve got to tell you that it is possible to do this exercise."},{"Start":"01:10.850 ","End":"01:13.430","Text":"In fact, more easily using conjugates,"},{"Start":"01:13.430 ","End":"01:15.740","Text":"because we have a square root plus something,"},{"Start":"01:15.740 ","End":"01:21.870","Text":"there is a technique, I think it was Chapter 2 of doing limits."},{"Start":"01:21.870 ","End":"01:24.930","Text":"Then the method of the conjugate was demonstrated."},{"Start":"01:24.930 ","End":"01:26.840","Text":"You can go ahead and do it with conjugates."},{"Start":"01:26.840 ","End":"01:28.625","Text":"But we\u0027re here to practice L\u0027Hopital,"},{"Start":"01:28.625 ","End":"01:31.820","Text":"even though it turns out slightly more difficult in this case."},{"Start":"01:31.820 ","End":"01:36.290","Text":"Before we can do L\u0027Hopital and L\u0027Hopital doesn\u0027t work for infinity minus infinity,"},{"Start":"01:36.290 ","End":"01:38.690","Text":"we\u0027ll have to do a bit of algebraic manipulation to"},{"Start":"01:38.690 ","End":"01:41.615","Text":"bring it to a form that is closer to L\u0027Hopital,"},{"Start":"01:41.615 ","End":"01:43.250","Text":"like 0 over 0,"},{"Start":"01:43.250 ","End":"01:46.550","Text":"infinity over infinity, or even 0 times infinity."},{"Start":"01:46.550 ","End":"01:50.390","Text":"We can get there. The algebra,"},{"Start":"01:50.390 ","End":"01:53.270","Text":"I first want to do is take x squared,"},{"Start":"01:53.270 ","End":"01:57.470","Text":"which is the exponent with the highest power outside the brackets under here."},{"Start":"01:57.470 ","End":"02:02.015","Text":"If I do that for this stuff here, take x squared out."},{"Start":"02:02.015 ","End":"02:03.905","Text":"This is what I\u0027m left with."},{"Start":"02:03.905 ","End":"02:08.030","Text":"The next thing I want to do is take the square root of x"},{"Start":"02:08.030 ","End":"02:12.275","Text":"squared to take the x squared outside the square root sign."},{"Start":"02:12.275 ","End":"02:15.080","Text":"It\u0027s not as immediate as you think,"},{"Start":"02:15.080 ","End":"02:18.140","Text":"you would say square root of x squared is x, but not always,"},{"Start":"02:18.140 ","End":"02:20.375","Text":"it\u0027s true for x positive, for x negative,"},{"Start":"02:20.375 ","End":"02:25.040","Text":"actually, the square root of x squared is minus x for negative x."},{"Start":"02:25.040 ","End":"02:27.815","Text":"You can try it substitute x equals minus 5,"},{"Start":"02:27.815 ","End":"02:30.380","Text":"and you\u0027ll see that you end up with plus 5,"},{"Start":"02:30.380 ","End":"02:32.750","Text":"which is actually minus x,"},{"Start":"02:32.750 ","End":"02:37.470","Text":"and so if we follow that,"},{"Start":"02:37.470 ","End":"02:42.230","Text":"then we have to take it outside the square root as minus x."},{"Start":"02:42.230 ","End":"02:46.370","Text":"The next thing to do is to see that we have x both here and here,"},{"Start":"02:46.370 ","End":"02:49.595","Text":"and take x as a common factor out."},{"Start":"02:49.595 ","End":"02:53.360","Text":"We end up with minus x times whatever was here,"},{"Start":"02:53.360 ","End":"02:56.660","Text":"but minus 1 because there was a minus."},{"Start":"02:56.660 ","End":"03:02.075","Text":"Now at this point, if you substitute x equals minus infinity,"},{"Start":"03:02.075 ","End":"03:04.340","Text":"minus infinity is infinity."},{"Start":"03:04.340 ","End":"03:06.230","Text":"Here we get 0 because like I said,"},{"Start":"03:06.230 ","End":"03:08.360","Text":"this thing goes to 0."},{"Start":"03:08.360 ","End":"03:10.495","Text":"We got square root of 1 minus 1."},{"Start":"03:10.495 ","End":"03:13.940","Text":"In other words, we\u0027re left with now infinity times 0."},{"Start":"03:13.940 ","End":"03:16.130","Text":"Now, infinity times 0 is very good."},{"Start":"03:16.130 ","End":"03:18.620","Text":"It\u0027s very close to L\u0027Hopital."},{"Start":"03:18.620 ","End":"03:23.485","Text":"Usually you either take the infinity part of the 0 part into the denominator."},{"Start":"03:23.485 ","End":"03:29.300","Text":"In this case, it\u0027s easier and more obvious to take the x part into the denominator."},{"Start":"03:29.300 ","End":"03:35.270","Text":"But remember when x goes into the denominator it goes as 1 over x."},{"Start":"03:35.270 ","End":"03:38.555","Text":"Just to show you an example and arithmetic of this,"},{"Start":"03:38.555 ","End":"03:40.010","Text":"suppose I had here,"},{"Start":"03:40.010 ","End":"03:43.970","Text":"I don\u0027t know, 10 times 4."},{"Start":"03:43.970 ","End":"03:47.330","Text":"I could put the first part into the denominator and"},{"Start":"03:47.330 ","End":"03:51.055","Text":"the reciprocal and say this equal to 4 over 1/10."},{"Start":"03:51.055 ","End":"03:53.450","Text":"If I wanted to put the 4 into the denominator,"},{"Start":"03:53.450 ","End":"03:56.015","Text":"I would say it\u0027s 10 over 1/4."},{"Start":"03:56.015 ","End":"03:58.490","Text":"This is basically what I\u0027m doing here when I put the x,"},{"Start":"03:58.490 ","End":"04:04.270","Text":"and you\u0027ll see this a lot things going from numerator to denominator and the 1 over."},{"Start":"04:06.980 ","End":"04:13.380","Text":"Having said that, we are about to do L\u0027Hopital for 0 over 0,"},{"Start":"04:13.820 ","End":"04:17.700","Text":"did I say this leaves us with 0 over 0, of course."},{"Start":"04:17.700 ","End":"04:21.945","Text":"This second part always was zero, and now the x,"},{"Start":"04:21.945 ","End":"04:24.555","Text":"which went to the infinity,"},{"Start":"04:24.555 ","End":"04:27.510","Text":"now, because it\u0027s 1 over x, it\u0027s a 0."},{"Start":"04:27.510 ","End":"04:30.550","Text":"What we\u0027re going to do is we\u0027re going to do L\u0027Hopital for 0 over 0."},{"Start":"04:30.550 ","End":"04:36.055","Text":"But before L\u0027Hopital, just in case you don\u0027t know all your formulae,"},{"Start":"04:36.055 ","End":"04:38.300","Text":"because we see a square root here,"},{"Start":"04:38.300 ","End":"04:40.325","Text":"we\u0027re going to have to differentiate."},{"Start":"04:40.325 ","End":"04:45.005","Text":"Just to remind you of this square root of something,"},{"Start":"04:45.005 ","End":"04:46.440","Text":"when you differentiate it,"},{"Start":"04:46.440 ","End":"04:49.400","Text":"it\u0027s 1 over twice the square root of that something times"},{"Start":"04:49.400 ","End":"04:53.390","Text":"the internal derivative from the chain rule."},{"Start":"04:54.500 ","End":"04:58.830","Text":"Having said this, we\u0027re now I\u0027m going to do a L\u0027Hopital."},{"Start":"04:58.830 ","End":"05:01.800","Text":"It was 0 over 0, we\u0027re applying L\u0027Hopital."},{"Start":"05:01.800 ","End":"05:04.595","Text":"L\u0027Hopital is said, instead of this limit, take a new limit,"},{"Start":"05:04.595 ","End":"05:08.695","Text":"which is what you get when you differentiate top and bottom separately."},{"Start":"05:08.695 ","End":"05:11.750","Text":"For the top using this formula, we get this,"},{"Start":"05:11.750 ","End":"05:16.225","Text":"and the internal derivative that\u0027s this part here is what\u0027s here,"},{"Start":"05:16.225 ","End":"05:18.425","Text":"and the derivative of this is this."},{"Start":"05:18.425 ","End":"05:21.575","Text":"Basically this is what we have now it looks horrible,"},{"Start":"05:21.575 ","End":"05:25.855","Text":"but really just need a bit of tidying up with algebra."},{"Start":"05:25.855 ","End":"05:29.315","Text":"But I see the page has come to an end,"},{"Start":"05:29.315 ","End":"05:35.580","Text":"so let\u0027s just copy it on to the next page."},{"Start":"05:38.900 ","End":"05:42.460","Text":"Starting a fresh page here."},{"Start":"05:44.300 ","End":"05:47.490","Text":"Now let\u0027s do some algebraic tidying."},{"Start":"05:47.490 ","End":"05:50.500","Text":"The stuff in the denominator goes to the denominator,"},{"Start":"05:50.500 ","End":"05:53.350","Text":"reciprocal can be inverted,"},{"Start":"05:53.350 ","End":"05:55.165","Text":"the x squared can come to the top."},{"Start":"05:55.165 ","End":"05:58.900","Text":"Basically, I\u0027m not going to go into the older minute algebraic details."},{"Start":"05:58.900 ","End":"06:04.930","Text":"We get this. Then we multiply this minus x squared by this expression."},{"Start":"06:04.930 ","End":"06:09.340","Text":"The minuses will cancel and the x squared will change the powers here."},{"Start":"06:09.340 ","End":"06:13.590","Text":"You can easily see that this is what we get."},{"Start":"06:13.590 ","End":"06:16.630","Text":"Now we\u0027re in a very good position because now we"},{"Start":"06:16.630 ","End":"06:19.784","Text":"can actually substitute the minus infinity."},{"Start":"06:19.784 ","End":"06:22.340","Text":"Now we substitute infinity or minus infinity,"},{"Start":"06:22.340 ","End":"06:24.694","Text":"1 over x is 0,"},{"Start":"06:24.694 ","End":"06:27.800","Text":"this is also 0, and this is also 0."},{"Start":"06:27.800 ","End":"06:30.110","Text":"If we put those 3 0s in,"},{"Start":"06:30.110 ","End":"06:32.300","Text":"then we get this expression."},{"Start":"06:32.300 ","End":"06:36.530","Text":"Of course we can throw the 0s out and the square root of 1 is 1,"},{"Start":"06:36.530 ","End":"06:41.430","Text":"and the final answer is minus 1/2, and we\u0027re done."}],"ID":1429}],"Thumbnail":null,"ID":52082}]

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