Introduction
0/1 completed

Technique 1 - Substitution
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Technique 2 - Factoring
0/5 completed

Technique 3 - Multiplying by the Conjugate
0/8 completed

Technique 4 - Function Tends to Infinity
0/12 completed

Technique 5 - X Tends to Infinity
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- X Tends to Infinity Part 1
- X Tends to Infinity Part 2
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10
- Exercise 11
- Exercise 12
- Exercise 13
- Exercise 14
- Exercise 15
- Exercise 16
- Exercise 17
- Exercise 18
- Exercise 19
- Exercise 20
- Exercise 21
- Exercise 22
- Exercise 23

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[{"Name":"Introduction","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Limit of a Function","Duration":"10m 15s","ChapterTopicVideoID":8248,"CourseChapterTopicPlaylistID":65358,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8248.jpeg","UploadDate":"2019-10-29T04:05:53.9970000","DurationForVideoObject":"PT10M15S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.380","Text":"In this clip, I\u0027ll be introducing the limit of a function."},{"Start":"00:04.380 ","End":"00:07.695","Text":"The concept of a limit is so important,"},{"Start":"00:07.695 ","End":"00:11.985","Text":"it\u0027s considered to be the basis of the differential and integral calculus."},{"Start":"00:11.985 ","End":"00:14.280","Text":"If you look ahead at the exercises,"},{"Start":"00:14.280 ","End":"00:17.310","Text":"you will see examples such as this,"},{"Start":"00:17.310 ","End":"00:19.980","Text":"which will look very strange to you if you don\u0027t know,"},{"Start":"00:19.980 ","End":"00:22.235","Text":"if you\u0027ve never seen this symbol."},{"Start":"00:22.235 ","End":"00:24.045","Text":"It\u0027s actually short for limit."},{"Start":"00:24.045 ","End":"00:28.170","Text":"In short, this will all look like gibberish and you won\u0027t know what\u0027s expected from you."},{"Start":"00:28.170 ","End":"00:31.935","Text":"Hopefully, by the end of this introduction, it\u0027ll be clearer."},{"Start":"00:31.935 ","End":"00:34.230","Text":"Meanwhile, let me get this out of your site."},{"Start":"00:34.230 ","End":"00:37.990","Text":"Let\u0027s begin by examining the function that appears here,"},{"Start":"00:37.990 ","End":"00:43.430","Text":"f of x is equal to x squared minus 1 over x minus 1."},{"Start":"00:43.430 ","End":"00:46.940","Text":"Its domain is clearly x not equal to"},{"Start":"00:46.940 ","End":"00:50.620","Text":"1 because that\u0027s the only value of x that could make it go wrong,"},{"Start":"00:50.620 ","End":"00:52.115","Text":"so let me write that."},{"Start":"00:52.115 ","End":"00:56.555","Text":"The domain is any x except x equals 1."},{"Start":"00:56.555 ","End":"00:58.580","Text":"If you try to substitute x equals 1,"},{"Start":"00:58.580 ","End":"01:01.430","Text":"you get 0 over 0 and that\u0027s not defined."},{"Start":"01:01.430 ","End":"01:03.905","Text":"But if I can\u0027t substitute x equals 1,"},{"Start":"01:03.905 ","End":"01:06.080","Text":"I could ask the following question."},{"Start":"01:06.080 ","End":"01:11.750","Text":"What happens to y or what does y approach when x approaches 1?"},{"Start":"01:11.750 ","End":"01:13.460","Text":"Let me try to explain."},{"Start":"01:13.460 ","End":"01:16.625","Text":"We\u0027re not allowed to substitute x equals 1."},{"Start":"01:16.625 ","End":"01:18.290","Text":"If I substitute x equals 1,"},{"Start":"01:18.290 ","End":"01:20.495","Text":"I don\u0027t have a value of y."},{"Start":"01:20.495 ","End":"01:24.200","Text":"But what if x just gets close to 1,"},{"Start":"01:24.200 ","End":"01:28.730","Text":"say x equals 0.99 or closer,"},{"Start":"01:28.730 ","End":"01:34.555","Text":"does y approach 4 maybe 10, 100 minus 1?"},{"Start":"01:34.555 ","End":"01:40.640","Text":"What happens here when x approaches a value where the function is undefined?"},{"Start":"01:40.640 ","End":"01:42.890","Text":"This end, we\u0027ll make a table of x,"},{"Start":"01:42.890 ","End":"01:46.235","Text":"y values and try to make an educated guess."},{"Start":"01:46.235 ","End":"01:47.765","Text":"Let\u0027s put some values in."},{"Start":"01:47.765 ","End":"01:50.250","Text":"We said x approaches 1."},{"Start":"01:50.250 ","End":"01:53.940","Text":"Let\u0027s begin with x equals 1.1."},{"Start":"01:53.940 ","End":"01:55.605","Text":"I\u0027ll be your calculator,"},{"Start":"01:55.605 ","End":"01:57.110","Text":"I\u0027ll substituted in here,"},{"Start":"01:57.110 ","End":"01:59.760","Text":"and y equals 2.1."},{"Start":"01:59.760 ","End":"02:03.245","Text":"Let\u0027s get even closer to 1, and let\u0027s try 1.01."},{"Start":"02:03.245 ","End":"02:06.260","Text":"Again. I\u0027ll be your calculator and I\u0027ll tell"},{"Start":"02:06.260 ","End":"02:09.740","Text":"you that what we get if we substitute 1.01 for x,"},{"Start":"02:09.740 ","End":"02:12.755","Text":"we\u0027ll get 2.01 for y."},{"Start":"02:12.755 ","End":"02:14.689","Text":"Let\u0027s get still closer,"},{"Start":"02:14.689 ","End":"02:21.195","Text":"1.001, you get 2.001."},{"Start":"02:21.195 ","End":"02:22.875","Text":"Let\u0027s do 1 more."},{"Start":"02:22.875 ","End":"02:26.355","Text":"When x equals 1.0001,"},{"Start":"02:26.355 ","End":"02:28.740","Text":"that\u0027s pretty close to 1,"},{"Start":"02:28.740 ","End":"02:33.180","Text":"do the computation, y equals 2.0001."},{"Start":"02:33.180 ","End":"02:39.830","Text":"All this leaves us with a distinct impression that when x approaches 1, y approaches 2."},{"Start":"02:39.830 ","End":"02:42.740","Text":"But something here is not quite right or not fair."},{"Start":"02:42.740 ","End":"02:45.380","Text":"All these values have been larger than 1."},{"Start":"02:45.380 ","End":"02:46.985","Text":"We\u0027re getting close to 1,"},{"Start":"02:46.985 ","End":"02:49.130","Text":"but we\u0027re staying above 1."},{"Start":"02:49.130 ","End":"02:51.500","Text":"We should really do it from the other side too,"},{"Start":"02:51.500 ","End":"02:54.920","Text":"and start with x equals say 0.9,"},{"Start":"02:54.920 ","End":"02:58.570","Text":"and approaches 1 from the other side. Let\u0027s do that."},{"Start":"02:58.570 ","End":"03:04.010","Text":"Let\u0027s start a second table and again we have x here and y here."},{"Start":"03:04.010 ","End":"03:07.610","Text":"This time we\u0027ll start from below 1 and work our way up."},{"Start":"03:07.610 ","End":"03:13.890","Text":"If x is 0.9, then y equals 1.9."},{"Start":"03:13.890 ","End":"03:16.235","Text":"That\u0027s for the rest, I\u0027ll just keep writing them in."},{"Start":"03:16.235 ","End":"03:22.510","Text":"If x equals 0.99, y equals 1.99."},{"Start":"03:22.510 ","End":"03:28.965","Text":"If we let x equals still closer 0.999, 1.999,"},{"Start":"03:28.965 ","End":"03:32.280","Text":"and the last 0.9999,"},{"Start":"03:32.280 ","End":"03:36.370","Text":"we get the value of y, 1.9999."},{"Start":"03:36.370 ","End":"03:40.790","Text":"We once again see that the values of y get very close to 2,"},{"Start":"03:40.790 ","End":"03:42.425","Text":"just as they did here."},{"Start":"03:42.425 ","End":"03:46.190","Text":"This allows us with some confidence to write the following."},{"Start":"03:46.190 ","End":"03:51.460","Text":"The Limit, which is written lim, just abbreviated,"},{"Start":"03:51.460 ","End":"04:00.605","Text":"as x goes to 1 of x squared minus 1 over x minus 1 is equal to 2."},{"Start":"04:00.605 ","End":"04:04.895","Text":"This is the notation lim for limit and this arrow,"},{"Start":"04:04.895 ","End":"04:06.230","Text":"x goes to 1,"},{"Start":"04:06.230 ","End":"04:08.930","Text":"x tends to 1, x approaches 1."},{"Start":"04:08.930 ","End":"04:10.730","Text":"That\u0027s the notation for limits."},{"Start":"04:10.730 ","End":"04:14.630","Text":"There\u0027s a comment I\u0027d like to make before we continue with the main theme."},{"Start":"04:14.630 ","End":"04:20.015","Text":"If I just want to describe what I concluded from this lower table,"},{"Start":"04:20.015 ","End":"04:25.100","Text":"where x approach 1 from below from numbers smaller than 1,"},{"Start":"04:25.100 ","End":"04:27.875","Text":"0.9999, and so on,"},{"Start":"04:27.875 ","End":"04:30.365","Text":"there is a special notation for that."},{"Start":"04:30.365 ","End":"04:38.510","Text":"We write x approaches 1 from the left and we write the little minus above the 1."},{"Start":"04:38.510 ","End":"04:41.145","Text":"This does not mean minus 1,"},{"Start":"04:41.145 ","End":"04:44.190","Text":"it just means slightly less than 1,"},{"Start":"04:44.190 ","End":"04:46.520","Text":"like 0.999 and so on."},{"Start":"04:46.520 ","End":"04:51.050","Text":"This describes this table and this sometimes we say the limit from the left."},{"Start":"04:51.050 ","End":"04:55.250","Text":"That the limit as x goes to 1 from the left of this and this is 2."},{"Start":"04:55.250 ","End":"04:58.010","Text":"Notice that I don\u0027t also put a minus above the 2,"},{"Start":"04:58.010 ","End":"05:00.485","Text":"even though it\u0027s slightly less than 2,"},{"Start":"05:00.485 ","End":"05:01.985","Text":"that\u0027s not of interest."},{"Start":"05:01.985 ","End":"05:04.805","Text":"Similarly, in the table above,"},{"Start":"05:04.805 ","End":"05:08.090","Text":"we had x approach 1 from numbers larger than 1,"},{"Start":"05:08.090 ","End":"05:10.820","Text":"1.1, 1.0, 1 etc."},{"Start":"05:10.820 ","End":"05:15.720","Text":"To describe this, we write a little plus above the 1,"},{"Start":"05:15.720 ","End":"05:19.340","Text":"and we say x approaches 1 from the right,"},{"Start":"05:19.340 ","End":"05:23.725","Text":"or the limit as x approaches 1 from the right of this function is 2."},{"Start":"05:23.725 ","End":"05:26.450","Text":"Again, we don\u0027t write that plus over the 2."},{"Start":"05:26.450 ","End":"05:29.840","Text":"These 2 are called the 1-sided limits,"},{"Start":"05:29.840 ","End":"05:32.900","Text":"the limit from the right and the limit from the left."},{"Start":"05:32.900 ","End":"05:35.060","Text":"We do have terms for these 2,"},{"Start":"05:35.060 ","End":"05:38.540","Text":"and this is just generally the limit from either side."},{"Start":"05:38.540 ","End":"05:43.190","Text":"Note that after I saw that the limit from above,"},{"Start":"05:43.190 ","End":"05:47.390","Text":"from the right was 2 and the limit from the left was 2,"},{"Start":"05:47.390 ","End":"05:50.810","Text":"then I had to write 2-sided limit as this is called"},{"Start":"05:50.810 ","End":"05:54.560","Text":"or just the plain limit was also equal to 2."},{"Start":"05:54.560 ","End":"05:57.050","Text":"Let\u0027s go on to the next page."},{"Start":"05:57.050 ","End":"06:00.020","Text":"To summarize this, I saw that limit from"},{"Start":"06:00.020 ","End":"06:03.300","Text":"the right was equal to the limit of the left which is equal to 2,"},{"Start":"06:03.300 ","End":"06:07.550","Text":"and then I wrote that the regular limit or 2-sided limit was equal to 2."},{"Start":"06:07.550 ","End":"06:10.535","Text":"Now, in most of our exercises,"},{"Start":"06:10.535 ","End":"06:13.190","Text":"we don\u0027t actually need to bother with limit"},{"Start":"06:13.190 ","End":"06:16.050","Text":"from the right and limit from the left and show that their equal."},{"Start":"06:16.050 ","End":"06:19.190","Text":"In most cases we just have techniques and we find the limit."},{"Start":"06:19.190 ","End":"06:22.250","Text":"There is 1 exceptional case where we will be dealing"},{"Start":"06:22.250 ","End":"06:25.600","Text":"with limit from the left separately and then add from the right separately,"},{"Start":"06:25.600 ","End":"06:27.980","Text":"and only when we conclude that these 2 are equal,"},{"Start":"06:27.980 ","End":"06:29.765","Text":"then we do the 2-sided limit."},{"Start":"06:29.765 ","End":"06:33.095","Text":"These are not so common, but you still have to know about this."},{"Start":"06:33.095 ","End":"06:35.990","Text":"What we learned here can actually be summarized"},{"Start":"06:35.990 ","End":"06:39.200","Text":"in a theorem which I\u0027m going to write below."},{"Start":"06:39.200 ","End":"06:42.360","Text":"Theorem, a function has a limit at a point,"},{"Start":"06:42.360 ","End":"06:43.460","Text":"and In our case,"},{"Start":"06:43.460 ","End":"06:46.280","Text":"this is the function and this is the point,"},{"Start":"06:46.280 ","End":"06:47.310","Text":"if and only if,"},{"Start":"06:47.310 ","End":"06:50.750","Text":"this is how in mathematics we say if and only if,"},{"Start":"06:50.750 ","End":"06:54.365","Text":"it has a limit from the left and from the right of the point,"},{"Start":"06:54.365 ","End":"06:56.530","Text":"and if these 2 are equal."},{"Start":"06:56.530 ","End":"06:59.090","Text":"Actually, I should have added something else that if"},{"Start":"06:59.090 ","End":"07:01.580","Text":"all of this happens and in this case,"},{"Start":"07:01.580 ","End":"07:03.830","Text":"that\u0027s the value of the limit of the function."},{"Start":"07:03.830 ","End":"07:06.950","Text":"For example, here this was 2 and this was 2,"},{"Start":"07:06.950 ","End":"07:08.705","Text":"so this limit not only exists,"},{"Start":"07:08.705 ","End":"07:10.115","Text":"but it\u0027s also equal to 2,"},{"Start":"07:10.115 ","End":"07:11.450","Text":"but I think that\u0027s implied."},{"Start":"07:11.450 ","End":"07:15.065","Text":"Before we continue to the actual techniques for finding limits,"},{"Start":"07:15.065 ","End":"07:19.495","Text":"I\u0027d like to illustrate all this graphically and they\u0027ll begin on the next page."},{"Start":"07:19.495 ","End":"07:22.325","Text":"I present the graph of our function."},{"Start":"07:22.325 ","End":"07:25.415","Text":"Note, especially this hole here."},{"Start":"07:25.415 ","End":"07:27.740","Text":"In general, this comes out to be a straight line,"},{"Start":"07:27.740 ","End":"07:29.360","Text":"but with a missing point,"},{"Start":"07:29.360 ","End":"07:32.150","Text":"because when x equals 1,"},{"Start":"07:32.150 ","End":"07:33.710","Text":"there is no value of y,"},{"Start":"07:33.710 ","End":"07:35.075","Text":"so I left a gap."},{"Start":"07:35.075 ","End":"07:41.865","Text":"But it certainly confirms the fact that when x is near 1 and y comes out near 2."},{"Start":"07:41.865 ","End":"07:46.370","Text":"What we did before was we just took values of x above 1,"},{"Start":"07:46.370 ","End":"07:52.460","Text":"getting closer, maybe some point here, and then here."},{"Start":"07:52.460 ","End":"07:55.890","Text":"Then we basically said, okay."},{"Start":"07:55.890 ","End":"08:00.065","Text":"That\u0027s x going to 1 from the right."},{"Start":"08:00.065 ","End":"08:02.780","Text":"Then we took points like the 0.99,"},{"Start":"08:02.780 ","End":"08:06.389","Text":"but various points that were on this side,"},{"Start":"08:06.550 ","End":"08:13.180","Text":"and this side, and that took care of the direction of coming to 1 from the left."},{"Start":"08:13.180 ","End":"08:19.395","Text":"This was the 1 minus and the other 1 was the 1 plus, let\u0027s call it."},{"Start":"08:19.395 ","End":"08:25.520","Text":"We computed the values of y and we got points on the graph when x is 1, there is no y."},{"Start":"08:25.520 ","End":"08:27.890","Text":"Nevertheless, if x is very close to 1,"},{"Start":"08:27.890 ","End":"08:32.000","Text":"y is very close to 2 and this line is 2 and this is 1."},{"Start":"08:32.000 ","End":"08:33.920","Text":"A big problem though,"},{"Start":"08:33.920 ","End":"08:37.010","Text":"is the fact that this is so informal, this process."},{"Start":"08:37.010 ","End":"08:39.050","Text":"What do I mean just taking a table,"},{"Start":"08:39.050 ","End":"08:40.370","Text":"putting in a few values,"},{"Start":"08:40.370 ","End":"08:42.200","Text":"and guessing what the trend is?"},{"Start":"08:42.200 ","End":"08:43.670","Text":"That just simply won\u0027t do."},{"Start":"08:43.670 ","End":"08:45.490","Text":"It\u0027s not mathematically enough."},{"Start":"08:45.490 ","End":"08:49.880","Text":"I\u0027d like to show you more algebraic way of getting to this value 2."},{"Start":"08:49.880 ","End":"08:53.600","Text":"I mean, we got ultimately that the conclusion that the limit was 2,"},{"Start":"08:53.600 ","End":"08:55.070","Text":"just from 2 tables."},{"Start":"08:55.070 ","End":"08:57.155","Text":"From the table, from the right table from the left,"},{"Start":"08:57.155 ","End":"08:59.810","Text":"there were equal, the numbers looked like they were getting close to 2."},{"Start":"08:59.810 ","End":"09:01.969","Text":"Now, let me do something more precise."},{"Start":"09:01.969 ","End":"09:04.310","Text":"Here\u0027s how I would do this algebraically."},{"Start":"09:04.310 ","End":"09:08.270","Text":"I would say the limit as x goes to 1,"},{"Start":"09:08.270 ","End":"09:09.845","Text":"this is the original question,"},{"Start":"09:09.845 ","End":"09:14.540","Text":"x squared minus 1 over x minus 1 is equal 2."},{"Start":"09:14.540 ","End":"09:18.755","Text":"What I would first of all do is some algebraic formula."},{"Start":"09:18.755 ","End":"09:21.310","Text":"But there\u0027s a thing called difference of squares formula."},{"Start":"09:21.310 ","End":"09:25.910","Text":"In any event, you must have seen this countless times that x squared minus 1 can be"},{"Start":"09:25.910 ","End":"09:32.695","Text":"factorized as 2 x minus 1 x plus 1 over x minus 1."},{"Start":"09:32.695 ","End":"09:36.140","Text":"It\u0027s agreed that when we take a limit where x tends to 1,"},{"Start":"09:36.140 ","End":"09:38.410","Text":"we do not include the value 1."},{"Start":"09:38.410 ","End":"09:40.565","Text":"We\u0027re not dividing by 0,"},{"Start":"09:40.565 ","End":"09:42.200","Text":"I mean x is from our domain."},{"Start":"09:42.200 ","End":"09:43.580","Text":"Is not equal to 1,"},{"Start":"09:43.580 ","End":"09:46.670","Text":"I mean, and x minus 1 is not 0."},{"Start":"09:46.670 ","End":"09:49.010","Text":"Therefore, just by the rules of fractions,"},{"Start":"09:49.010 ","End":"09:50.960","Text":"were allowed to cancel something in"},{"Start":"09:50.960 ","End":"09:53.855","Text":"the numerator and in the denominator or a common factor."},{"Start":"09:53.855 ","End":"09:56.370","Text":"What we\u0027re left here is just x plus 1."},{"Start":"09:56.370 ","End":"10:02.440","Text":"Now 1 of the first techniques we\u0027ll learn in the limits is method of substitution."},{"Start":"10:02.440 ","End":"10:05.720","Text":"If there\u0027s no particular problem and x plus 1 is not problematic,"},{"Start":"10:05.720 ","End":"10:07.430","Text":"we just substitute the value."},{"Start":"10:07.430 ","End":"10:10.340","Text":"You just put x equals 1 into here."},{"Start":"10:10.340 ","End":"10:11.870","Text":"We got to the answer 2,"},{"Start":"10:11.870 ","End":"10:13.430","Text":"which we were expecting."},{"Start":"10:13.430 ","End":"10:16.140","Text":"This will do as an introduction."}],"ID":8408}],"Thumbnail":null,"ID":65358},{"Name":"Technique 1 - Substitution","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Substitution","Duration":"8m 43s","ChapterTopicVideoID":2,"CourseChapterTopicPlaylistID":65359,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/2.jpeg","UploadDate":"2019-10-29T04:06:42.3630000","DurationForVideoObject":"PT8M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"Several techniques for computing the limit of a function."},{"Start":"00:03.570 ","End":"00:06.645","Text":"In this clip we\u0027ll be d1emonstrating the first 1,"},{"Start":"00:06.645 ","End":"00:09.565","Text":"which is known as substitution."},{"Start":"00:09.565 ","End":"00:16.530","Text":"What this basically says is that if the function is defined at the given point,"},{"Start":"00:16.530 ","End":"00:19.890","Text":"which was not the case that we had before in"},{"Start":"00:19.890 ","End":"00:24.345","Text":"the example with x squared minus 1 over x minus 1."},{"Start":"00:24.345 ","End":"00:27.090","Text":"It leaves the function is defined at a given point that in"},{"Start":"00:27.090 ","End":"00:30.105","Text":"order to compute the limit of the function at this point,"},{"Start":"00:30.105 ","End":"00:33.945","Text":"we simply substitute the point in the function."},{"Start":"00:33.945 ","End":"00:36.810","Text":"I\u0027m going to show you a few examples."},{"Start":"00:36.810 ","End":"00:41.250","Text":"Then I\u0027m going to tell you that this is not quite true, but don\u0027t worry."},{"Start":"00:41.410 ","End":"00:45.740","Text":"Let\u0027s take here for examples that should do for a start,"},{"Start":"00:45.740 ","End":"00:51.755","Text":"the limit as x tends to 1 of x squared minus 4x plus 1."},{"Start":"00:51.755 ","End":"00:58.010","Text":"Now note that this function is defined for all values of x."},{"Start":"00:58.010 ","End":"01:02.780","Text":"In particular, it\u0027s defined for x is 1."},{"Start":"01:02.780 ","End":"01:05.780","Text":"In fact, in all these examples that I\u0027ve given here,"},{"Start":"01:05.780 ","End":"01:10.000","Text":"these functions are always defined at the limit value."},{"Start":"01:10.000 ","End":"01:11.795","Text":"In this case, in each,"},{"Start":"01:11.795 ","End":"01:14.240","Text":"we can use substitution and all of them."},{"Start":"01:14.240 ","End":"01:16.280","Text":"What do we do for this 1,"},{"Start":"01:16.280 ","End":"01:19.160","Text":"for example, x goes to 1."},{"Start":"01:19.160 ","End":"01:22.340","Text":"I\u0027m going to substitute instead of x,"},{"Start":"01:22.340 ","End":"01:24.840","Text":"I\u0027m going to put 1."},{"Start":"01:24.890 ","End":"01:27.620","Text":"Here I\u0027ve done that substitution."},{"Start":"01:27.620 ","End":"01:29.510","Text":"I\u0027ve put 1 instead of x,"},{"Start":"01:29.510 ","End":"01:34.475","Text":"so we get 1 squared minus 4 times 1 plus 1."},{"Start":"01:34.475 ","End":"01:43.655","Text":"That\u0027s equal to minus 2 because we have 1 minus 4 plus 1, that\u0027s minus 2."},{"Start":"01:43.655 ","End":"01:46.645","Text":"That\u0027s the answer to the first one."},{"Start":"01:46.645 ","End":"01:48.700","Text":"Now in the second one,"},{"Start":"01:48.700 ","End":"01:51.280","Text":"all I have to do is substitute x equals 4,"},{"Start":"01:51.280 ","End":"01:55.385","Text":"and we\u0027ll do that in blue in this exercise,"},{"Start":"01:55.385 ","End":"01:59.250","Text":"and we\u0027ll get this."},{"Start":"01:59.250 ","End":"02:01.445","Text":"If we compute this,"},{"Start":"02:01.445 ","End":"02:03.275","Text":"and we can do this in our heads,"},{"Start":"02:03.275 ","End":"02:08.340","Text":"a 104 minus 4 is a 100 and the square root of a 100 is 10."},{"Start":"02:09.710 ","End":"02:12.240","Text":"Yet another example."},{"Start":"02:12.240 ","End":"02:16.795","Text":"First thing is to substitute x equals 10."},{"Start":"02:16.795 ","End":"02:20.990","Text":"Notice that we substituting 10 here,"},{"Start":"02:20.990 ","End":"02:25.940","Text":"but we could have substituted anything that is anything except minus 18."},{"Start":"02:25.940 ","End":"02:29.960","Text":"This is one of those functions which is not defined everywhere,"},{"Start":"02:29.960 ","End":"02:32.265","Text":"but it is our point which is 10."},{"Start":"02:32.265 ","End":"02:40.205","Text":"We get this that x we put 10 in blue even."},{"Start":"02:40.205 ","End":"02:42.080","Text":"We compute this."},{"Start":"02:42.080 ","End":"02:44.840","Text":"10 plus 4 is 14."},{"Start":"02:44.840 ","End":"02:46.490","Text":"10 plus 18 is 28,"},{"Start":"02:46.490 ","End":"02:50.920","Text":"14 over 28 is equal to 1.5."},{"Start":"02:50.920 ","End":"02:52.845","Text":"That\u0027s the answer to this 1."},{"Start":"02:52.845 ","End":"02:54.860","Text":"The last one\u0027s a bit of a funny one."},{"Start":"02:54.860 ","End":"02:58.310","Text":"People sometimes confused by the constant function."},{"Start":"02:58.310 ","End":"03:00.290","Text":"This is the constant function 40,"},{"Start":"03:00.290 ","End":"03:03.220","Text":"which means for any x is 40."},{"Start":"03:03.220 ","End":"03:05.180","Text":"It doesn\u0027t care about X,"},{"Start":"03:05.180 ","End":"03:07.550","Text":"whatever x is, it\u0027s 40."},{"Start":"03:07.550 ","End":"03:12.425","Text":"The limit as x goes to a 140 is still 40."},{"Start":"03:12.425 ","End":"03:14.880","Text":"Those who don\u0027t quite get it,"},{"Start":"03:14.880 ","End":"03:16.310","Text":"sometimes I had an idea,"},{"Start":"03:16.310 ","End":"03:19.659","Text":"wants to explain 40 instead of 40,"},{"Start":"03:19.659 ","End":"03:24.855","Text":"I wrote it as 40 plus 0 x."},{"Start":"03:24.855 ","End":"03:27.739","Text":"Everyone was happy that x is in the function."},{"Start":"03:27.739 ","End":"03:30.130","Text":"Substitute x equals a 100,"},{"Start":"03:30.130 ","End":"03:32.940","Text":"40 plus 0 times a 100 is 40."},{"Start":"03:32.940 ","End":"03:36.285","Text":"If you\u0027re happier that way, It\u0027s okay."},{"Start":"03:36.285 ","End":"03:39.860","Text":"Up till now, we\u0027ve defined what to do,"},{"Start":"03:39.860 ","End":"03:41.435","Text":"and we\u0027ve given some examples."},{"Start":"03:41.435 ","End":"03:45.350","Text":"The only thing is that what I\u0027ve written here is not exactly true."},{"Start":"03:45.350 ","End":"03:49.990","Text":"There\u0027s a bit of some exceptions and let\u0027s take care of those now."},{"Start":"03:49.990 ","End":"03:53.545","Text":"What it says that this technique, substitution,"},{"Start":"03:53.545 ","End":"03:59.014","Text":"is valid as long as the function is defined by 1 single formula."},{"Start":"03:59.014 ","End":"04:01.220","Text":"Just like the functions in high school"},{"Start":"04:01.220 ","End":"04:06.040","Text":"where in the case it\u0027s defined by 2 or more formally,"},{"Start":"04:06.040 ","End":"04:12.025","Text":"we proceed differently exactly how we shall see in what is to come."},{"Start":"04:12.025 ","End":"04:19.720","Text":"Not at this point. I just wanted to mention that it doesn\u0027t always work the substitution."},{"Start":"04:20.330 ","End":"04:26.915","Text":"The other thing I\u0027d like to raise and people ask is,"},{"Start":"04:26.915 ","End":"04:30.910","Text":"if the substitution is just so basic, so simple,"},{"Start":"04:30.910 ","End":"04:37.175","Text":"why would anyone ever ask you to solve a limit with just a substitution?"},{"Start":"04:37.175 ","End":"04:41.110","Text":"The answer is, it usually won\u0027t happen unless you\u0027re very lucky,"},{"Start":"04:41.110 ","End":"04:43.015","Text":"you won\u0027t get such a thing in an exam."},{"Start":"04:43.015 ","End":"04:46.985","Text":"However, in the course of solving a question on limits,"},{"Start":"04:46.985 ","End":"04:49.790","Text":"maybe you\u0027d be doing some simplifications and that"},{"Start":"04:49.790 ","End":"04:56.270","Text":"changing and processing during the course of it you will get such thing."},{"Start":"04:56.270 ","End":"04:58.895","Text":"He may simplify something more complicated"},{"Start":"04:58.895 ","End":"05:01.805","Text":"to something which is just needs a substitution."},{"Start":"05:01.805 ","End":"05:07.600","Text":"It is definitely a useful technique to learn."},{"Start":"05:08.840 ","End":"05:13.819","Text":"In this exception paragraph,"},{"Start":"05:13.819 ","End":"05:18.290","Text":"I talked about single formula and multiple formulae."},{"Start":"05:18.290 ","End":"05:20.360","Text":"You might say, what does that mean?"},{"Start":"05:20.360 ","End":"05:24.570","Text":"Well, I\u0027ll illustrate in the case of an example,"},{"Start":"05:26.540 ","End":"05:36.890","Text":"so it could be defined in it cooled a piece wise fashion or in a split form."},{"Start":"05:36.890 ","End":"05:42.740","Text":"We say, the value of f of x depends on what we\u0027re x is."},{"Start":"05:42.740 ","End":"05:45.500","Text":"For x, which are bigger or equal to 5,"},{"Start":"05:45.500 ","End":"05:46.835","Text":"we define it 1 way,"},{"Start":"05:46.835 ","End":"05:48.665","Text":"namely x squared plus 1."},{"Start":"05:48.665 ","End":"05:50.990","Text":"But if X is less than 5,"},{"Start":"05:50.990 ","End":"05:52.534","Text":"then we use the other formula,"},{"Start":"05:52.534 ","End":"05:53.990","Text":"minus x plus 7."},{"Start":"05:53.990 ","End":"05:57.320","Text":"For example, if I want to know what f of 6 is,"},{"Start":"05:57.320 ","End":"05:59.555","Text":"6 is bigger or equal to 5."},{"Start":"05:59.555 ","End":"06:03.385","Text":"The answer is 6, 6 squared plus 1 or 37."},{"Start":"06:03.385 ","End":"06:06.110","Text":"F of 6 is 37. On the other hand,"},{"Start":"06:06.110 ","End":"06:08.795","Text":"f of 2, I put 2 here."},{"Start":"06:08.795 ","End":"06:11.765","Text":"Let\u0027s use this formula because 2 is smaller than 5."},{"Start":"06:11.765 ","End":"06:14.945","Text":"It\u0027s minus 2 plus 7, so it\u0027s 5."},{"Start":"06:14.945 ","End":"06:19.205","Text":"Now, the problem occurs when we have the limit."},{"Start":"06:19.205 ","End":"06:21.515","Text":"Something happens around 5."},{"Start":"06:21.515 ","End":"06:25.760","Text":"It\u0027s maybe a scene point,"},{"Start":"06:25.760 ","End":"06:28.775","Text":"seam line or transition point."},{"Start":"06:28.775 ","End":"06:30.530","Text":"If I was to ask,"},{"Start":"06:30.530 ","End":"06:39.650","Text":"what is the limit as x goes to 5 of f of x?"},{"Start":"06:39.650 ","End":"06:45.680","Text":"Then we can\u0027t use the technique of substitution because 5"},{"Start":"06:45.680 ","End":"06:51.520","Text":"is on a seam line between different areas with different formulae."},{"Start":"06:51.520 ","End":"06:59.590","Text":"That was the exception to assume we have 2 formulae and not 1."},{"Start":"07:01.690 ","End":"07:06.260","Text":"In any event, most functions are defined by"},{"Start":"07:06.260 ","End":"07:10.340","Text":"a simple formula that takes care of most of the cases."},{"Start":"07:10.340 ","End":"07:16.205","Text":"But yes, we will see later on split or piecewise-defined functions."},{"Start":"07:16.205 ","End":"07:18.275","Text":"We\u0027re basically done."},{"Start":"07:18.275 ","End":"07:22.160","Text":"In fact, we are done for those of you who wanted to be done,"},{"Start":"07:22.160 ","End":"07:26.340","Text":"but I have to add that there\u0027s still an exception to the exception."},{"Start":"07:27.640 ","End":"07:30.620","Text":"But this is also not entirely true."},{"Start":"07:30.620 ","End":"07:34.775","Text":"There are very, what we might call exotic kinds of a function."},{"Start":"07:34.775 ","End":"07:38.180","Text":"Now, you don\u0027t have to listen any further unless you\u0027re interested in."},{"Start":"07:38.180 ","End":"07:42.080","Text":"But we have very weird functions or maybe put it,"},{"Start":"07:42.080 ","End":"07:43.340","Text":"put it in a different color."},{"Start":"07:43.340 ","End":"07:46.055","Text":"Say, I don\u0027t know in the green,"},{"Start":"07:46.055 ","End":"07:48.695","Text":"it can define functions in any ways."},{"Start":"07:48.695 ","End":"07:54.815","Text":"You could define, let\u0027s say f of x is equal to positive 1."},{"Start":"07:54.815 ","End":"07:58.530","Text":"If x is a rational number,"},{"Start":"08:00.680 ","End":"08:06.830","Text":"and 0 if x is irrational in the mathematical sense,"},{"Start":"08:06.830 ","End":"08:09.120","Text":"meaning it\u0027s not a fraction."},{"Start":"08:12.020 ","End":"08:15.605","Text":"Even this is not really defined."},{"Start":"08:15.605 ","End":"08:18.050","Text":"It\u0027s defined only in terms of 2 formulae,"},{"Start":"08:18.050 ","End":"08:25.680","Text":"but it\u0027ll be entirely different than the techniques."},{"Start":"08:26.630 ","End":"08:29.200","Text":"It\u0027s just something else,"},{"Start":"08:29.200 ","End":"08:32.465","Text":"and they\u0027re a very bizarre things and functions that you might see."},{"Start":"08:32.465 ","End":"08:36.110","Text":"Anyway, that\u0027s just beyond the schedule at this point."},{"Start":"08:36.110 ","End":"08:38.150","Text":"You won\u0027t see them in Calculus 1."},{"Start":"08:38.150 ","End":"08:42.090","Text":"That\u0027s reassuring. We\u0027re done for now."}],"ID":2},{"Watched":false,"Name":"Exercise 1","Duration":"46s","ChapterTopicVideoID":1524,"CourseChapterTopicPlaylistID":65359,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1524.jpeg","UploadDate":"2014-10-22T03:56:40.1570000","DurationForVideoObject":"PT46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.020","Text":"In this exercise, we have to find the limit as x"},{"Start":"00:04.020 ","End":"00:08.670","Text":"tends to 4 of the function x squared plus x plus 1."},{"Start":"00:08.670 ","End":"00:10.965","Text":"This is an elementary function,"},{"Start":"00:10.965 ","End":"00:13.920","Text":"and so if it\u0027s defined at x equals 4,"},{"Start":"00:13.920 ","End":"00:19.920","Text":"which it is, all we have to do is substitute x equals 4 in the function."},{"Start":"00:19.920 ","End":"00:25.980","Text":"What we get is that the limit as x tends to"},{"Start":"00:25.980 ","End":"00:32.820","Text":"4 of x squared plus x plus 1 equals,"},{"Start":"00:32.820 ","End":"00:35.250","Text":"I\u0027m going to substitute 4 for x,"},{"Start":"00:35.250 ","End":"00:39.765","Text":"is 4 squared plus 4 plus 1;"},{"Start":"00:39.765 ","End":"00:44.520","Text":"16 plus 4 plus 1 gives us 21,"},{"Start":"00:44.520 ","End":"00:46.960","Text":"and that\u0027s the answer."}],"ID":1536},{"Watched":false,"Name":"Exercise 2","Duration":"47s","ChapterTopicVideoID":1525,"CourseChapterTopicPlaylistID":65359,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1525.jpeg","UploadDate":"2014-10-22T03:56:46.8730000","DurationForVideoObject":"PT47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.390","Text":"In this exercise, we have to find the limit as x"},{"Start":"00:03.390 ","End":"00:07.800","Text":"tends to 10 of the function x plus 1 over x plus 2."},{"Start":"00:07.800 ","End":"00:12.930","Text":"This is an elementary function which means that if we can substitute x equals 10,"},{"Start":"00:12.930 ","End":"00:15.150","Text":"that\u0027s also going to be the limit."},{"Start":"00:15.150 ","End":"00:18.135","Text":"Let\u0027s see if we can substitute x equals 10,"},{"Start":"00:18.135 ","End":"00:23.370","Text":"rewrite the exercise limit as x tends to 10"},{"Start":"00:23.370 ","End":"00:30.165","Text":"of x plus 1 over x plus 2 is equal."},{"Start":"00:30.165 ","End":"00:32.775","Text":"Instead of x I\u0027m going to put 10."},{"Start":"00:32.775 ","End":"00:39.315","Text":"I get 10 plus 1 over 10 plus 2."},{"Start":"00:39.315 ","End":"00:44.655","Text":"That\u0027s equal to 11 over 12."},{"Start":"00:44.655 ","End":"00:48.330","Text":"That\u0027s the answer to all there is to it."}],"ID":1537},{"Watched":false,"Name":"Exercise 3","Duration":"1m 3s","ChapterTopicVideoID":1526,"CourseChapterTopicPlaylistID":65359,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1526.jpeg","UploadDate":"2014-10-22T03:56:51.0470000","DurationForVideoObject":"PT1M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.930","Text":"In this exercise, we have to find the limit as x tends to"},{"Start":"00:03.930 ","End":"00:09.150","Text":"1 of the function square root of x plus 3. Not quite."},{"Start":"00:09.150 ","End":"00:11.355","Text":"Notice that there\u0027s a little plus here."},{"Start":"00:11.355 ","End":"00:14.520","Text":"This means the limit as x goes to 1 from the right."},{"Start":"00:14.520 ","End":"00:16.200","Text":"But I\u0027ll ignore that for the moment,"},{"Start":"00:16.200 ","End":"00:18.385","Text":"and take care of that at the end."},{"Start":"00:18.385 ","End":"00:21.150","Text":"This function is an elementary function,"},{"Start":"00:21.150 ","End":"00:24.495","Text":"which means that if we can substitute x equals 1,"},{"Start":"00:24.495 ","End":"00:26.460","Text":"that will also be the limit."},{"Start":"00:26.460 ","End":"00:31.205","Text":"There\u0027s no reason why we can\u0027t substitute x equals 1 here, so let\u0027s see."},{"Start":"00:31.205 ","End":"00:38.360","Text":"I\u0027ll just put instead of x_1 as the square root of 1 plus 3 is 2."},{"Start":"00:38.360 ","End":"00:40.490","Text":"Okay, that\u0027s the regular limit."},{"Start":"00:40.490 ","End":"00:44.570","Text":"But in theory, you learned that if a function has a regular limit,"},{"Start":"00:44.570 ","End":"00:46.490","Text":"sometimes called a 2-sided limit,"},{"Start":"00:46.490 ","End":"00:50.690","Text":"and it also has a limit from the left and a limit from the right and all are equal."},{"Start":"00:50.690 ","End":"00:52.730","Text":"Just to make it precise,"},{"Start":"00:52.730 ","End":"00:57.770","Text":"I can also say that the limit as x tends to 1"},{"Start":"00:57.770 ","End":"01:04.350","Text":"from the right is also equal to 2, and we\u0027re done."}],"ID":1538},{"Watched":false,"Name":"Exercise 4","Duration":"37s","ChapterTopicVideoID":1527,"CourseChapterTopicPlaylistID":65359,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1527.jpeg","UploadDate":"2014-10-22T03:56:53.5700000","DurationForVideoObject":"PT37S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.175","Text":"In this exercise, we have to find the limit as x tends to 100 of the function 20."},{"Start":"00:05.175 ","End":"00:06.930","Text":"This is a constant function."},{"Start":"00:06.930 ","End":"00:08.610","Text":"It\u0027s an elementary function,"},{"Start":"00:08.610 ","End":"00:10.635","Text":"and certainly we can substitute x equals"},{"Start":"00:10.635 ","End":"00:14.460","Text":"100 and it\u0027ll just give us 20 and that will be the limit."},{"Start":"00:14.460 ","End":"00:23.130","Text":"In other words, the limit as x tends to 100"},{"Start":"00:23.130 ","End":"00:32.475","Text":"of 20 is equal to the same expression where I substitute x is 100,"},{"Start":"00:32.475 ","End":"00:35.400","Text":"well the 20 doesn\u0027t care, it\u0027s just 20."},{"Start":"00:35.400 ","End":"00:38.530","Text":"That\u0027s the answer. We\u0027re done."}],"ID":1539}],"Thumbnail":null,"ID":65359},{"Name":"Technique 2 - Factoring","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Factoring","Duration":"6m 28s","ChapterTopicVideoID":8249,"CourseChapterTopicPlaylistID":65360,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8249.jpeg","UploadDate":"2019-10-29T04:06:46.8170000","DurationForVideoObject":"PT6M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.185","Text":"In this clip, we talk about technique number 2 for computing a limit."},{"Start":"00:04.185 ","End":"00:07.200","Text":"It\u0027s called factoring and/or canceling."},{"Start":"00:07.200 ","End":"00:10.049","Text":"If the limit is of the form 0 over 0,"},{"Start":"00:10.049 ","End":"00:14.340","Text":"we try to break the numerator or the denominator or both,"},{"Start":"00:14.340 ","End":"00:16.110","Text":"and the factors, hopefully,"},{"Start":"00:16.110 ","End":"00:18.810","Text":"they have some common factor which we can cancel."},{"Start":"00:18.810 ","End":"00:20.970","Text":"I\u0027ll illustrate with a few examples."},{"Start":"00:20.970 ","End":"00:22.995","Text":"Let\u0027s take the first example,"},{"Start":"00:22.995 ","End":"00:30.840","Text":"the limit as x goes to 1 of x squared minus 1 over x minus 1."},{"Start":"00:30.840 ","End":"00:34.925","Text":"The first thing to do in the limit is to try the substitution method."},{"Start":"00:34.925 ","End":"00:36.710","Text":"If I put x equals 1,"},{"Start":"00:36.710 ","End":"00:38.880","Text":"then x minus 1 is 0,"},{"Start":"00:38.880 ","End":"00:41.415","Text":"x squared minus 1 is also 0,"},{"Start":"00:41.415 ","End":"00:44.625","Text":"so we do indeed have a 0 over 0 case."},{"Start":"00:44.625 ","End":"00:47.105","Text":"In the numerator, we have x squared minus 1,"},{"Start":"00:47.105 ","End":"00:51.620","Text":"and that\u0027s easily factored using the difference of squares formula."},{"Start":"00:51.620 ","End":"00:55.770","Text":"We end up getting the limit as x goes to 1,"},{"Start":"00:55.770 ","End":"01:00.425","Text":"this thing becomes x plus 1, x minus 1,"},{"Start":"01:00.425 ","End":"01:03.140","Text":"and in the denominator there\u0027s nothing to factorize,"},{"Start":"01:03.140 ","End":"01:05.620","Text":"so leave it as x minus 1."},{"Start":"01:05.620 ","End":"01:08.420","Text":"This point, we noticed that something can be canceled,"},{"Start":"01:08.420 ","End":"01:11.075","Text":"x minus 1 cancels with x minus 1,"},{"Start":"01:11.075 ","End":"01:17.495","Text":"and all we\u0027re left with is the limit as x goes to 1 of x plus 1,"},{"Start":"01:17.495 ","End":"01:20.420","Text":"and here, the substitution method will work."},{"Start":"01:20.420 ","End":"01:22.325","Text":"Just put x equals 1 here,"},{"Start":"01:22.325 ","End":"01:23.810","Text":"the answer is 2."},{"Start":"01:23.810 ","End":"01:28.875","Text":"Next example, limit as x goes to 0,"},{"Start":"01:28.875 ","End":"01:35.695","Text":"x squared plus 4x over x squared minus 10x."},{"Start":"01:35.695 ","End":"01:38.855","Text":"As before, I try putting x equals 0,"},{"Start":"01:38.855 ","End":"01:41.115","Text":"x squared minus 10x comes out a 0."},{"Start":"01:41.115 ","End":"01:44.270","Text":"Basically, the x being 0 makes all the terms 0,"},{"Start":"01:44.270 ","End":"01:48.085","Text":"and we end up again with the 0 over 0 form."},{"Start":"01:48.085 ","End":"01:51.350","Text":"We try the method of factoring and canceling,"},{"Start":"01:51.350 ","End":"01:53.810","Text":"and we get the limit again,"},{"Start":"01:53.810 ","End":"01:55.250","Text":"x goes to 0."},{"Start":"01:55.250 ","End":"01:58.580","Text":"The x squared plus 4x becomes x plus 4."},{"Start":"01:58.580 ","End":"02:02.930","Text":"We just take the x outside the brackets and on the denominator,"},{"Start":"02:02.930 ","End":"02:05.570","Text":"x times x minus 10."},{"Start":"02:05.570 ","End":"02:07.100","Text":"Next step is to cancel."},{"Start":"02:07.100 ","End":"02:09.020","Text":"We see that x is common,"},{"Start":"02:09.020 ","End":"02:11.405","Text":"and I should note that x is not 0."},{"Start":"02:11.405 ","End":"02:14.615","Text":"You might think I\u0027m canceling by 0 because when x tends to 0,"},{"Start":"02:14.615 ","End":"02:16.190","Text":"x does not equal 0,"},{"Start":"02:16.190 ","End":"02:17.660","Text":"so this is okay,"},{"Start":"02:17.660 ","End":"02:24.950","Text":"and we end up with the limit as x goes to 0 of x plus 4 over x minus 10."},{"Start":"02:24.950 ","End":"02:28.090","Text":"At this point, there\u0027s no problem in substitution,"},{"Start":"02:28.090 ","End":"02:29.600","Text":"so let x equals 0,"},{"Start":"02:29.600 ","End":"02:33.260","Text":"and end up with 4 over minus 10."},{"Start":"02:33.260 ","End":"02:34.759","Text":"For those who like decimals,"},{"Start":"02:34.759 ","End":"02:37.810","Text":"I could write this as minus 0.4,"},{"Start":"02:37.810 ","End":"02:39.565","Text":"and that\u0027s this example."},{"Start":"02:39.565 ","End":"02:42.670","Text":"The next exercise is a fake exercise,"},{"Start":"02:42.670 ","End":"02:44.590","Text":"and I\u0027ll explain what I mean in a minute."},{"Start":"02:44.590 ","End":"02:50.955","Text":"Limit as x goes to 1 of twice x minus 1 times x"},{"Start":"02:50.955 ","End":"02:58.485","Text":"plus 4 over 5 times x minus 1 times x plus 7."},{"Start":"02:58.485 ","End":"03:00.495","Text":"What did I mean by fake exercise?"},{"Start":"03:00.495 ","End":"03:01.735","Text":"I mean it\u0027s too easy."},{"Start":"03:01.735 ","End":"03:03.925","Text":"Look, it\u0027s already been factored for you."},{"Start":"03:03.925 ","End":"03:06.205","Text":"You can already see that it has the common factor."},{"Start":"03:06.205 ","End":"03:08.740","Text":"Here, first you check that it\u0027s a 0 over 0,"},{"Start":"03:08.740 ","End":"03:09.850","Text":"you put x equals 1,"},{"Start":"03:09.850 ","End":"03:11.995","Text":"here 0, here 0, and so on."},{"Start":"03:11.995 ","End":"03:14.200","Text":"Then you\u0027d say my work\u0027s all cut out for me,"},{"Start":"03:14.200 ","End":"03:16.090","Text":"x minus 1 is a common factor."},{"Start":"03:16.090 ","End":"03:19.755","Text":"Cancel it, then substitute x equals 1,"},{"Start":"03:19.755 ","End":"03:21.090","Text":"and at this point,"},{"Start":"03:21.090 ","End":"03:25.370","Text":"you would just do some simple arithmetic and everything falls into place."},{"Start":"03:25.370 ","End":"03:26.570","Text":"Even the 5 cancels,"},{"Start":"03:26.570 ","End":"03:28.745","Text":"2 over 8 leaves you with 1/4."},{"Start":"03:28.745 ","End":"03:30.230","Text":"This is unrealistic,"},{"Start":"03:30.230 ","End":"03:31.310","Text":"as I said, because it is too easy."},{"Start":"03:31.310 ","End":"03:34.310","Text":"What\u0027s more likely is that the professor or whoever"},{"Start":"03:34.310 ","End":"03:37.595","Text":"made up the exercise wrote something like this for himself,"},{"Start":"03:37.595 ","End":"03:41.105","Text":"but then he multiplied it out to make it more difficult on you."},{"Start":"03:41.105 ","End":"03:43.670","Text":"He would give you this exercise,"},{"Start":"03:43.670 ","End":"03:45.410","Text":"limit as x goes to 1,"},{"Start":"03:45.410 ","End":"03:55.425","Text":"2x squared plus 6x minus 8 over 5x squared plus 30x minus 35,"},{"Start":"03:55.425 ","End":"03:58.760","Text":"and you somehow have to factor it to get to here."},{"Start":"03:58.760 ","End":"04:00.020","Text":"Here in the frame,"},{"Start":"04:00.020 ","End":"04:05.405","Text":"I\u0027ve written the technique that you use to factorize a quadratic polynomial,"},{"Start":"04:05.405 ","End":"04:11.240","Text":"ax squared plus bx plus c. What you do is you first of all go to the next line,"},{"Start":"04:11.240 ","End":"04:13.370","Text":"which is to solve the quadratic equation."},{"Start":"04:13.370 ","End":"04:14.900","Text":"You set it equal to 0,"},{"Start":"04:14.900 ","End":"04:17.660","Text":"you solve the equation with the famous formula,"},{"Start":"04:17.660 ","End":"04:20.530","Text":"I\u0027m not going to repeat it, and you get 2x\u0027s."},{"Start":"04:20.530 ","End":"04:22.080","Text":"Then you get these 2x\u0027s,"},{"Start":"04:22.080 ","End":"04:23.715","Text":"which I call x_1 and x_2,"},{"Start":"04:23.715 ","End":"04:25.575","Text":"stick them in this formula,"},{"Start":"04:25.575 ","End":"04:27.320","Text":"write this thing as a,"},{"Start":"04:27.320 ","End":"04:30.140","Text":"which is the coefficient of x squared, and then in brackets,"},{"Start":"04:30.140 ","End":"04:34.245","Text":"x minus 1 of the roots times x minus the other root."},{"Start":"04:34.245 ","End":"04:37.175","Text":"Let\u0027s apply the technique from the box to here."},{"Start":"04:37.175 ","End":"04:39.800","Text":"We have twice the quadratic polynomial,"},{"Start":"04:39.800 ","End":"04:41.445","Text":"so let\u0027s write it as a fraction."},{"Start":"04:41.445 ","End":"04:48.260","Text":"This 1, we can write according to this rule as 2 times x minus the x_1 there,"},{"Start":"04:48.260 ","End":"04:50.740","Text":"x minus the x_2 there,"},{"Start":"04:50.740 ","End":"04:52.885","Text":"where the 2 is this 2,"},{"Start":"04:52.885 ","End":"04:55.425","Text":"and the same thing on the denominator,"},{"Start":"04:55.425 ","End":"05:01.160","Text":"5x minus something, x minus something,"},{"Start":"05:01.160 ","End":"05:04.295","Text":"where the 5 here and here are the same."},{"Start":"05:04.295 ","End":"05:06.470","Text":"The next thing to do, and I\u0027m not going to do,"},{"Start":"05:06.470 ","End":"05:08.750","Text":"it is to solve the quadratic equations."},{"Start":"05:08.750 ","End":"05:13.505","Text":"I solve this thing equals 0 and I\u0027ll get 2 solutions,"},{"Start":"05:13.505 ","End":"05:15.275","Text":"1 and minus 4,"},{"Start":"05:15.275 ","End":"05:17.165","Text":"but you\u0027ll do that on your own."},{"Start":"05:17.165 ","End":"05:20.360","Text":"The same here, you solve this thing equals 0."},{"Start":"05:20.360 ","End":"05:22.235","Text":"You get the 2 solutions for x,"},{"Start":"05:22.235 ","End":"05:24.305","Text":"a 1 and minus 7,"},{"Start":"05:24.305 ","End":"05:26.690","Text":"and then at the end for the factorization,"},{"Start":"05:26.690 ","End":"05:29.045","Text":"we put the x_1 x_2 here."},{"Start":"05:29.045 ","End":"05:30.755","Text":"Here\u0027s x minus 1,"},{"Start":"05:30.755 ","End":"05:33.420","Text":"x minus minus minus 4,"},{"Start":"05:33.420 ","End":"05:35.685","Text":"and here, x minus 1,"},{"Start":"05:35.685 ","End":"05:38.010","Text":"x minus minus 7."},{"Start":"05:38.010 ","End":"05:41.310","Text":"But you should replace these minus minus,"},{"Start":"05:41.310 ","End":"05:43.110","Text":"you put it as a plus, of course."},{"Start":"05:43.110 ","End":"05:47.000","Text":"All that remains to be done is to continue from this point."},{"Start":"05:47.000 ","End":"05:50.225","Text":"This is this and we just continue, it\u0027s not the answer."},{"Start":"05:50.225 ","End":"05:53.090","Text":"As you see, it\u0027s quite a lot of work to get from"},{"Start":"05:53.090 ","End":"05:56.485","Text":"this limit here all the way to the answer,"},{"Start":"05:56.485 ","End":"06:01.400","Text":"and there\u0027s extra work that I didn\u0027t do of solving 2 quadratic equations."},{"Start":"06:01.400 ","End":"06:04.220","Text":"But I have some good news that in the future,"},{"Start":"06:04.220 ","End":"06:07.925","Text":"you\u0027ll be learning a new rule that will help enormously."},{"Start":"06:07.925 ","End":"06:11.270","Text":"Later, you\u0027ll be learning something called L\u0027Hopital\u0027s rule,"},{"Start":"06:11.270 ","End":"06:16.670","Text":"which is really magical and will make all this work happen in a matter of seconds."},{"Start":"06:16.670 ","End":"06:18.290","Text":"It will come later, but first,"},{"Start":"06:18.290 ","End":"06:20.890","Text":"you\u0027ll have to learn differentiation."},{"Start":"06:20.890 ","End":"06:22.590","Text":"Until such time,"},{"Start":"06:22.590 ","End":"06:23.930","Text":"you have to do it the hard way,"},{"Start":"06:23.930 ","End":"06:26.420","Text":"but you\u0027ll know the shortcut rule is on the way."},{"Start":"06:26.420 ","End":"06:29.880","Text":"That\u0027s all I have to say for this clip."}],"ID":8409},{"Watched":false,"Name":"Exercise 1","Duration":"3m 41s","ChapterTopicVideoID":1528,"CourseChapterTopicPlaylistID":65360,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1528.jpeg","UploadDate":"2014-10-22T04:00:31.0170000","DurationForVideoObject":"PT3M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.230","Text":"In this exercise, we have to find the limit as x tends to 3."},{"Start":"00:04.230 ","End":"00:08.504","Text":"Now, it\u0027s an elementary function and if we\u0027re lucky,"},{"Start":"00:08.504 ","End":"00:10.500","Text":"and it\u0027s defined where x equals 3,"},{"Start":"00:10.500 ","End":"00:12.885","Text":"we just have to substitute x equals 3."},{"Start":"00:12.885 ","End":"00:15.210","Text":"Unfortunately, this is not the case."},{"Start":"00:15.210 ","End":"00:18.165","Text":"We put x equals 3 here,"},{"Start":"00:18.165 ","End":"00:25.725","Text":"we\u0027ll get that x squared minus 9 is 3 squared minus 9 is 0."},{"Start":"00:25.725 ","End":"00:27.390","Text":"A denominator of 0,"},{"Start":"00:27.390 ","End":"00:32.430","Text":"so 3 is not defined here so the trick of substitution won\u0027t work."},{"Start":"00:32.430 ","End":"00:34.315","Text":"We\u0027ll have to try something else."},{"Start":"00:34.315 ","End":"00:37.730","Text":"The standard trick to be used in cases like this"},{"Start":"00:37.730 ","End":"00:41.480","Text":"is to factorize both numerator and denominator."},{"Start":"00:41.480 ","End":"00:44.090","Text":"Let\u0027s start with the denominator."},{"Start":"00:44.090 ","End":"00:49.925","Text":"It\u0027s easier. We have x squared minus 9 equals."},{"Start":"00:49.925 ","End":"00:57.230","Text":"Now, let me remind you of a basic formula in algebra, difference of squares,"},{"Start":"00:57.230 ","End":"01:07.455","Text":"that a squared minus b squared is equal to a minus b, a plus b."},{"Start":"01:07.455 ","End":"01:09.795","Text":"If we apply it in our case,"},{"Start":"01:09.795 ","End":"01:11.745","Text":"9 is 3 squared."},{"Start":"01:11.745 ","End":"01:16.960","Text":"We get x minus 3, x plus 3."},{"Start":"01:16.960 ","End":"01:18.895","Text":"For the numerator,"},{"Start":"01:18.895 ","End":"01:23.915","Text":"I\u0027m going to have to give you another formula that when we have an expression,"},{"Start":"01:23.915 ","End":"01:29.630","Text":"x squared plus ax plus b,"},{"Start":"01:29.630 ","End":"01:39.365","Text":"this is equal to x minus x_1 times x minus x_2,"},{"Start":"01:39.365 ","End":"01:47.090","Text":"where x_1 and x_2 are the roots of the quadratic expression."},{"Start":"01:47.090 ","End":"01:51.260","Text":"In other words, the solutions to the equation where this is equal to 0."},{"Start":"01:51.260 ","End":"02:00.490","Text":"In our numerator, which is x squared minus x minus 6, to find the roots,"},{"Start":"02:00.490 ","End":"02:04.635","Text":"we solve the equation equals 0."},{"Start":"02:04.635 ","End":"02:08.690","Text":"Since you know how to solve the quadratic equations,"},{"Start":"02:08.690 ","End":"02:11.585","Text":"I\u0027ll just give you the answer straightaway."},{"Start":"02:11.585 ","End":"02:14.765","Text":"So applying what we wrote here,"},{"Start":"02:14.765 ","End":"02:21.740","Text":"we can say that x squared minus x minus 6 is"},{"Start":"02:21.740 ","End":"02:29.360","Text":"equal to x minus the first root times x minus the second root,"},{"Start":"02:29.360 ","End":"02:31.220","Text":"which makes it a plus,"},{"Start":"02:31.220 ","End":"02:34.205","Text":"and that factorizes this."},{"Start":"02:34.205 ","End":"02:36.335","Text":"Scroll down a bit."},{"Start":"02:36.335 ","End":"02:46.970","Text":"What we have if we write the original expression is the limit as x tends to 3."},{"Start":"02:46.970 ","End":"02:53.105","Text":"The numerator, which we factorized to be x minus 3,"},{"Start":"02:53.105 ","End":"02:56.900","Text":"x plus 2, and the denominator,"},{"Start":"02:56.900 ","End":"03:02.555","Text":"x minus 3 times x plus 3."},{"Start":"03:02.555 ","End":"03:05.615","Text":"Now, notice when x tends to 3,"},{"Start":"03:05.615 ","End":"03:07.580","Text":"it\u0027s not equal to 3."},{"Start":"03:07.580 ","End":"03:11.360","Text":"So x minus 3 is not 0 and we can cancel."},{"Start":"03:11.360 ","End":"03:16.550","Text":"At this point we have the limit as x goes to 3 of a new function,"},{"Start":"03:16.550 ","End":"03:19.085","Text":"x plus 2 over x plus 3."},{"Start":"03:19.085 ","End":"03:24.560","Text":"It\u0027s also elementary and here we can substitute x equals 3."},{"Start":"03:24.560 ","End":"03:27.290","Text":"So this is just equal to,"},{"Start":"03:27.290 ","End":"03:34.475","Text":"we get 3 plus 2/3 plus 3,"},{"Start":"03:34.475 ","End":"03:42.820","Text":"and that\u0027s just 5/6 and that\u0027s the answer. We\u0027re done."}],"ID":1540},{"Watched":false,"Name":"Exercise 2","Duration":"5m 5s","ChapterTopicVideoID":1529,"CourseChapterTopicPlaylistID":65360,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1529.jpeg","UploadDate":"2014-10-22T04:00:53.2970000","DurationForVideoObject":"PT5M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.065","Text":"In this exercise, we have to find the limit as x goes to minus 5 of this function,"},{"Start":"00:07.065 ","End":"00:11.640","Text":"2x squared minus 50 over 2x squared plus 3x minus 35."},{"Start":"00:11.640 ","End":"00:13.860","Text":"This is an elementary function."},{"Start":"00:13.860 ","End":"00:18.345","Text":"If we\u0027re lucky enough that it\u0027s defined for minus 5,"},{"Start":"00:18.345 ","End":"00:21.505","Text":"then all we have to do is substitute minus 5."},{"Start":"00:21.505 ","End":"00:24.510","Text":"But unfortunately that\u0027s not so if you do"},{"Start":"00:24.510 ","End":"00:30.065","Text":"a mental computation of putting minus 5 for x in the denominator,"},{"Start":"00:30.065 ","End":"00:36.860","Text":"minus 5 squared is 25 times 2 is 50 minus 15 minus 35, we get a 0."},{"Start":"00:36.860 ","End":"00:40.460","Text":"In this case, we have to use the technique"},{"Start":"00:40.460 ","End":"00:44.285","Text":"of factorization of the numerator and the denominator."},{"Start":"00:44.285 ","End":"00:46.865","Text":"Let\u0027s start with the numerator."},{"Start":"00:46.865 ","End":"00:49.370","Text":"Here I wrote the formula I see we\u0027re going to need."},{"Start":"00:49.370 ","End":"00:59.390","Text":"Factorizing the numerator, we get 2x squared minus 50 is equal."},{"Start":"00:59.390 ","End":"01:02.030","Text":"First of all, let\u0027s take 2 outside the brackets,"},{"Start":"01:02.030 ","End":"01:10.440","Text":"2x squared minus 25 and 25 is 5 squared."},{"Start":"01:10.440 ","End":"01:12.200","Text":"If I use this formula,"},{"Start":"01:12.200 ","End":"01:14.420","Text":"which is the difference of squares formula,"},{"Start":"01:14.420 ","End":"01:23.505","Text":"we\u0027ll get that this is equal to 2 x minus 5, x plus 5."},{"Start":"01:23.505 ","End":"01:27.335","Text":"That\u0027s as far as the numerator can be simplified."},{"Start":"01:27.335 ","End":"01:29.095","Text":"Now the denominator."},{"Start":"01:29.095 ","End":"01:37.380","Text":"Here we have 2x squared plus 3x minus 35."},{"Start":"01:37.380 ","End":"01:41.170","Text":"I\u0027m going to need another formula for you."},{"Start":"01:41.170 ","End":"01:43.800","Text":"This is the formula I wanted to use,"},{"Start":"01:43.800 ","End":"01:46.130","Text":"that the quadratic expression,"},{"Start":"01:46.130 ","End":"01:50.995","Text":"x squared plus bx plus c is equal to a times x minus x_1,"},{"Start":"01:50.995 ","End":"01:52.890","Text":"x minus x_2,"},{"Start":"01:52.890 ","End":"01:58.070","Text":"and x_1 and x_2 are the roots of this quadratic expression,"},{"Start":"01:58.070 ","End":"02:02.650","Text":"which means the solution of the equation where all this is equal to 0."},{"Start":"02:02.650 ","End":"02:06.065","Text":"What we have to do first is find those roots."},{"Start":"02:06.065 ","End":"02:12.125","Text":"Let\u0027s just write here equal to 0 and we\u0027ll solve this equation."},{"Start":"02:12.125 ","End":"02:16.745","Text":"I\u0027m assuming that you know how to solve quadratic equations."},{"Start":"02:16.745 ","End":"02:18.965","Text":"I\u0027ll just tell you the answer."},{"Start":"02:18.965 ","End":"02:20.225","Text":"What this means,"},{"Start":"02:20.225 ","End":"02:22.025","Text":"according to this formula,"},{"Start":"02:22.025 ","End":"02:27.320","Text":"is that we can rewrite what was the denominator as 2x squared"},{"Start":"02:27.320 ","End":"02:35.210","Text":"plus 3x minus 35 is equal to 2,"},{"Start":"02:35.210 ","End":"02:37.865","Text":"that\u0027s the a part,"},{"Start":"02:37.865 ","End":"02:41.480","Text":"times x minus the first root,"},{"Start":"02:41.480 ","End":"02:43.820","Text":"7 over 2,"},{"Start":"02:43.820 ","End":"02:46.295","Text":"times x minus the other root,"},{"Start":"02:46.295 ","End":"02:50.165","Text":"x minus 5, let\u0027s just write it as plus 5."},{"Start":"02:50.165 ","End":"02:53.585","Text":"If I multiply the first brackets by 2,"},{"Start":"02:53.585 ","End":"02:55.835","Text":"it\u0027ll be a little bit simpler."},{"Start":"02:55.835 ","End":"02:58.835","Text":"I\u0027ll get 2x minus 7,"},{"Start":"02:58.835 ","End":"03:02.520","Text":"x plus 5 without fractions."},{"Start":"03:02.520 ","End":"03:07.640","Text":"Now what I\u0027m going to want to do is to substitute the numerator over"},{"Start":"03:07.640 ","End":"03:12.320","Text":"the denominator here in terms of the new factoring we found."},{"Start":"03:12.320 ","End":"03:18.620","Text":"I\u0027ll scroll down a bit and we can rewrite our exercise."},{"Start":"03:18.620 ","End":"03:23.895","Text":"The limit of 2x squared minus 50"},{"Start":"03:23.895 ","End":"03:31.485","Text":"over 2x squared plus 3x minus 35."},{"Start":"03:31.485 ","End":"03:34.310","Text":"Of course, I have to say where the limit is."},{"Start":"03:34.310 ","End":"03:38.450","Text":"It\u0027s where x tends to minus 5."},{"Start":"03:38.450 ","End":"03:43.670","Text":"This is equal to the limit of what we factored these as."},{"Start":"03:43.670 ","End":"03:51.980","Text":"The first 1, the numerator is twice x minus 5, x plus 5."},{"Start":"03:51.980 ","End":"03:55.475","Text":"The denominator, that\u0027s the work we did here,"},{"Start":"03:55.475 ","End":"04:01.085","Text":"is 2x minus 7 times x"},{"Start":"04:01.085 ","End":"04:07.725","Text":"plus 5 as x tends to minus 5."},{"Start":"04:07.725 ","End":"04:09.560","Text":"Now looking at this,"},{"Start":"04:09.560 ","End":"04:13.145","Text":"we have an x plus 5 in the numerator and in the denominator."},{"Start":"04:13.145 ","End":"04:15.590","Text":"Now, x is not equal to minus 5,"},{"Start":"04:15.590 ","End":"04:17.635","Text":"it only tends to minus 5,"},{"Start":"04:17.635 ","End":"04:19.485","Text":"so this thing is not 0."},{"Start":"04:19.485 ","End":"04:22.025","Text":"We can do this big cancellation."},{"Start":"04:22.025 ","End":"04:27.360","Text":"That really helps us because what remains is still an elementary expression,"},{"Start":"04:27.360 ","End":"04:31.545","Text":"only this time we can substitute minus 5 in it."},{"Start":"04:31.545 ","End":"04:34.460","Text":"The limit is just a substitution."},{"Start":"04:34.460 ","End":"04:37.640","Text":"We get twice minus 5,"},{"Start":"04:37.640 ","End":"04:46.720","Text":"minus 5 over twice minus 5, minus 7."},{"Start":"04:46.720 ","End":"04:49.940","Text":"The numerator is twice minus 10,"},{"Start":"04:49.940 ","End":"04:52.070","Text":"which is minus 20."},{"Start":"04:52.070 ","End":"04:57.650","Text":"The denominator minus 10 minus 7 is minus 17."},{"Start":"04:57.650 ","End":"05:03.020","Text":"The answer is 20 over 17."},{"Start":"05:03.020 ","End":"05:06.490","Text":"This is it and we\u0027re done."}],"ID":1541},{"Watched":false,"Name":"Exercise 3","Duration":"3m 8s","ChapterTopicVideoID":1530,"CourseChapterTopicPlaylistID":65360,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1530.jpeg","UploadDate":"2014-10-22T04:01:07.1070000","DurationForVideoObject":"PT3M8S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.170","Text":"In this exercise, we have to find the limit when x tends to"},{"Start":"00:04.170 ","End":"00:10.020","Text":"1 of the function x^7 minus x over x minus 1."},{"Start":"00:10.020 ","End":"00:12.255","Text":"It\u0027s an elementary function."},{"Start":"00:12.255 ","End":"00:14.905","Text":"If we could substitute x equals 1,"},{"Start":"00:14.905 ","End":"00:16.310","Text":"that would give us the answer."},{"Start":"00:16.310 ","End":"00:19.220","Text":"Unfortunately, we can\u0027t substitute x equals 1"},{"Start":"00:19.220 ","End":"00:22.730","Text":"because of the denominator that would make it 0."},{"Start":"00:22.730 ","End":"00:25.790","Text":"We have to use the technique of factorization."},{"Start":"00:25.790 ","End":"00:28.250","Text":"We factorize a numerator and denominator,"},{"Start":"00:28.250 ","End":"00:30.400","Text":"and hopefully something cancels."},{"Start":"00:30.400 ","End":"00:32.700","Text":"The denominator doesn\u0027t need to be factored."},{"Start":"00:32.700 ","End":"00:40.400","Text":"Let\u0027s factorize the numerator and let\u0027s see what is x^7 minus x, what this equals."},{"Start":"00:40.400 ","End":"00:42.410","Text":"I have to show you a formula."},{"Start":"00:42.410 ","End":"00:45.395","Text":"Here is the formula and I\u0027ll return to that when I need it."},{"Start":"00:45.395 ","End":"00:48.820","Text":"In our case, we first factor out an x."},{"Start":"00:48.820 ","End":"00:55.900","Text":"We get x, x times x^6 minus 1."},{"Start":"00:55.900 ","End":"01:01.520","Text":"Now, I\u0027m going to use the formula over here,"},{"Start":"01:01.520 ","End":"01:05.990","Text":"this time with n equals 6."},{"Start":"01:05.990 ","End":"01:09.190","Text":"This will give us that if this is 6,"},{"Start":"01:09.190 ","End":"01:15.520","Text":"here we\u0027ll have a^5th plus a^4,"},{"Start":"01:15.520 ","End":"01:18.890","Text":"plus a^3 like cubed,"},{"Start":"01:18.890 ","End":"01:24.455","Text":"plus a squared plus a plus 1."},{"Start":"01:24.455 ","End":"01:26.660","Text":"That\u0027s just for this bit here,"},{"Start":"01:26.660 ","End":"01:30.755","Text":"cause we still have the a minus 1."},{"Start":"01:30.755 ","End":"01:35.160","Text":"In our case, x is the role of a,"},{"Start":"01:35.160 ","End":"01:45.100","Text":"so we get this thing equals x times the x minus 1 times this thing,"},{"Start":"01:45.100 ","End":"01:51.695","Text":"but with x instead of a. X^5, plus x^4,"},{"Start":"01:51.695 ","End":"01:54.075","Text":"plus x cubed,"},{"Start":"01:54.075 ","End":"01:56.310","Text":"plus x squared,"},{"Start":"01:56.310 ","End":"02:00.075","Text":"plus x plus 1."},{"Start":"02:00.075 ","End":"02:04.560","Text":"Now, we have to rewrite this limit."},{"Start":"02:04.560 ","End":"02:15.025","Text":"What we\u0027re looking for is the limit as x tends to or goes to 1 of x^7 minus x,"},{"Start":"02:15.025 ","End":"02:16.075","Text":"which is all this."},{"Start":"02:16.075 ","End":"02:20.035","Text":"I\u0027ll just copy it out again quickly,"},{"Start":"02:20.035 ","End":"02:25.430","Text":"over the original x minus 1."},{"Start":"02:25.430 ","End":"02:27.770","Text":"Now, here we\u0027re lucky."},{"Start":"02:27.770 ","End":"02:30.725","Text":"We can cancel the x minus 1."},{"Start":"02:30.725 ","End":"02:34.670","Text":"Notice that this is not 0 because when x tends to 1,"},{"Start":"02:34.670 ","End":"02:36.410","Text":"it\u0027s not equal to 1."},{"Start":"02:36.410 ","End":"02:41.060","Text":"What\u0027s left is a polynomial an elementary function."},{"Start":"02:41.060 ","End":"02:45.620","Text":"To find the limit we just have to substitute 1 instead of x."},{"Start":"02:45.620 ","End":"02:53.960","Text":"This is equal to 1 times 1^5, plus 1^4,"},{"Start":"02:53.960 ","End":"02:57.265","Text":"plus 1 cubed,"},{"Start":"02:57.265 ","End":"03:04.665","Text":"plus 1 squared plus 1 and another 1 is 6."},{"Start":"03:04.665 ","End":"03:09.850","Text":"So that\u0027s the answer to the exercise. We\u0027re done."}],"ID":1542},{"Watched":false,"Name":"Exercise 4","Duration":"4m 21s","ChapterTopicVideoID":1531,"CourseChapterTopicPlaylistID":65360,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1531.jpeg","UploadDate":"2014-10-22T04:01:26.2130000","DurationForVideoObject":"PT4M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.780","Text":"In this exercise, we have to find the limit as x tends to"},{"Start":"00:03.780 ","End":"00:09.990","Text":"1 of x to the n minus x over x minus 1."},{"Start":"00:09.990 ","End":"00:14.550","Text":"I should add that we have to assume that n is some positive whole number,"},{"Start":"00:14.550 ","End":"00:19.470","Text":"preferably bigger than 1 and integer."},{"Start":"00:19.470 ","End":"00:25.800","Text":"This function x to the n minus x over x minus 1 is an elementary function."},{"Start":"00:25.800 ","End":"00:29.505","Text":"If we could substitute x equals 1,"},{"Start":"00:29.505 ","End":"00:30.705","Text":"if that were allowed,"},{"Start":"00:30.705 ","End":"00:32.955","Text":"that would just give us our answer."},{"Start":"00:32.955 ","End":"00:37.170","Text":"But we can\u0027t because if we put x equals 1 in the denominator,"},{"Start":"00:37.170 ","End":"00:38.820","Text":"we see that we get 0."},{"Start":"00:38.820 ","End":"00:45.649","Text":"We\u0027ll have to use a technique of factoring numerator and denominator. Let\u0027s begin."},{"Start":"00:45.649 ","End":"00:55.069","Text":"The limit as x tends to 1 of x to the n minus x"},{"Start":"00:55.069 ","End":"01:05.505","Text":"over x minus 1 is equal to the limit as x goes to 1."},{"Start":"01:05.505 ","End":"01:09.140","Text":"We can take x out as a factor from the numerator,"},{"Start":"01:09.140 ","End":"01:20.070","Text":"x times x to the n minus 1 minus 1 over x minus 1."},{"Start":"01:20.070 ","End":"01:23.750","Text":"We have to know what to do with this term here."},{"Start":"01:23.750 ","End":"01:27.120","Text":"Let me remind you of a formula."},{"Start":"01:27.170 ","End":"01:29.995","Text":"This is the formula I mean."},{"Start":"01:29.995 ","End":"01:35.150","Text":"I\u0027ve deliberately chosen other letters than x and n to make it general,"},{"Start":"01:35.150 ","End":"01:37.100","Text":"that a to the power of k,"},{"Start":"01:37.100 ","End":"01:38.615","Text":"where k is a natural number,"},{"Start":"01:38.615 ","End":"01:44.900","Text":"minus 1 is a minus 1 times the sum of descending powers of a,"},{"Start":"01:44.900 ","End":"01:46.895","Text":"a to the k minus 1,"},{"Start":"01:46.895 ","End":"01:51.140","Text":"a to the k minus 2 and so on plus a plus 1."},{"Start":"01:51.140 ","End":"01:54.890","Text":"If we apply that here, continuing,"},{"Start":"01:54.890 ","End":"01:58.775","Text":"we get that this is equal to"},{"Start":"01:58.775 ","End":"02:05.930","Text":"the limit as x goes to 1 of x in the numerator."},{"Start":"02:05.930 ","End":"02:12.690","Text":"Now, this thing, it\u0027s like k is n minus 1 and a is x."},{"Start":"02:12.860 ","End":"02:17.280","Text":"The a minus 1 is the x minus 1."},{"Start":"02:17.280 ","End":"02:20.635","Text":"This sum begins from k minus 1."},{"Start":"02:20.635 ","End":"02:24.310","Text":"In other words, it begins from n minus 2,"},{"Start":"02:24.310 ","End":"02:34.600","Text":"so it\u0027s x to the n minus 2 plus x to the n minus 3 and so on plus x plus 1."},{"Start":"02:34.600 ","End":"02:40.445","Text":"The denominator is the same old denominator x minus 1."},{"Start":"02:40.445 ","End":"02:46.610","Text":"But the x minus 1 is in the numerator and in the denominator,"},{"Start":"02:46.610 ","End":"02:49.180","Text":"so we can cancel it."},{"Start":"02:49.180 ","End":"02:53.700","Text":"It\u0027s not 0 because when x tends to 1, it isn\u0027t 1."},{"Start":"02:53.700 ","End":"03:00.045","Text":"What we\u0027re left with is the limit as x goes to 1 of"},{"Start":"03:00.045 ","End":"03:07.245","Text":"x times x to the n minus 2 plus and so on,"},{"Start":"03:07.245 ","End":"03:12.585","Text":"plus x plus 1 over 1."},{"Start":"03:12.585 ","End":"03:18.050","Text":"This equals, all I have to do is substitute instead of x, 1,"},{"Start":"03:18.050 ","End":"03:24.205","Text":"and I get 1 times 1 plus dot dot dot,"},{"Start":"03:24.205 ","End":"03:27.390","Text":"plus 1, plus 1."},{"Start":"03:27.390 ","End":"03:31.880","Text":"Now, the question is how many 1s are there?"},{"Start":"03:31.880 ","End":"03:34.045","Text":"Well, if you notice,"},{"Start":"03:34.045 ","End":"03:38.335","Text":"we begin with this 1 which is x to the power of 0,"},{"Start":"03:38.335 ","End":"03:40.075","Text":"then x to the power of 1."},{"Start":"03:40.075 ","End":"03:42.490","Text":"In other words, the powers go,"},{"Start":"03:42.490 ","End":"03:44.710","Text":"if I take it from left to right,"},{"Start":"03:44.710 ","End":"03:46.930","Text":"the powers are 0,"},{"Start":"03:46.930 ","End":"03:52.005","Text":"1 and so on up to n minus 2."},{"Start":"03:52.005 ","End":"03:57.085","Text":"Now, if I take all the numbers from 0 up to a certain number,"},{"Start":"03:57.085 ","End":"04:01.960","Text":"then there\u0027s going to be a total of n minus 1 of them."},{"Start":"04:01.960 ","End":"04:03.910","Text":"If it was from 0 to,"},{"Start":"04:03.910 ","End":"04:05.425","Text":"let\u0027s say, 8,"},{"Start":"04:05.425 ","End":"04:07.885","Text":"there\u0027d be 9 of them and so on."},{"Start":"04:07.885 ","End":"04:12.075","Text":"What we have is n minus 1, 1s."},{"Start":"04:12.075 ","End":"04:15.435","Text":"1 plus 1 plus 1 plus 1 n minus 1 times."},{"Start":"04:15.435 ","End":"04:22.180","Text":"That just gives us n minus 1 and that\u0027s our answer. We\u0027re done."}],"ID":1543}],"Thumbnail":null,"ID":65360},{"Name":"Technique 3 - Multiplying by the Conjugate","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Multiplying by The Conjugate","Duration":"8m 13s","ChapterTopicVideoID":8250,"CourseChapterTopicPlaylistID":65361,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8250.jpeg","UploadDate":"2019-10-29T04:06:57.8630000","DurationForVideoObject":"PT8M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.900","Text":"In this clip, we\u0027re going to talk about technique number 3,"},{"Start":"00:03.900 ","End":"00:08.910","Text":"and it\u0027s called multiplying by the conjugate or multiplication with the conjugate."},{"Start":"00:08.910 ","End":"00:12.555","Text":"First, I\u0027ll tell you what the indications that we should use it."},{"Start":"00:12.555 ","End":"00:14.655","Text":"Well, there\u0027s 2 main things."},{"Start":"00:14.655 ","End":"00:19.070","Text":"1 is that the limit is of the form 0/0."},{"Start":"00:19.070 ","End":"00:20.600","Text":"It means when you substitute the value,"},{"Start":"00:20.600 ","End":"00:23.285","Text":"that\u0027s what it is, that\u0027s what you get."},{"Start":"00:23.285 ","End":"00:27.060","Text":"The second thing is that there"},{"Start":"00:27.060 ","End":"00:37.065","Text":"is a square root sign,"},{"Start":"00:37.065 ","End":"00:39.380","Text":"could be numerator or denominator."},{"Start":"00:39.380 ","End":"00:40.985","Text":"Let\u0027s start with an example."},{"Start":"00:40.985 ","End":"00:46.550","Text":"The example will be limit as x goes to 1"},{"Start":"00:46.550 ","End":"00:53.360","Text":"of the square root of 15 plus x minus 4,"},{"Start":"00:53.360 ","End":"00:58.305","Text":"all over x minus 1."},{"Start":"00:58.305 ","End":"01:01.625","Text":"Let\u0027s see, do these conditions hold?"},{"Start":"01:01.625 ","End":"01:03.470","Text":"Well, if I put x as 1,"},{"Start":"01:03.470 ","End":"01:06.095","Text":"15 plus x is 16, square root of 16,"},{"Start":"01:06.095 ","End":"01:08.740","Text":"4 minus 4 is 0,"},{"Start":"01:08.740 ","End":"01:11.700","Text":"and 1 minus 1 is 0."},{"Start":"01:11.700 ","End":"01:18.500","Text":"We do indeed have the 0/0 form."},{"Start":"01:18.500 ","End":"01:23.330","Text":"Secondly, you want to be looking at the numerator and/or the denominator,"},{"Start":"01:23.330 ","End":"01:25.055","Text":"and we do have a square root."},{"Start":"01:25.055 ","End":"01:27.520","Text":"There is the square root."},{"Start":"01:27.980 ","End":"01:30.765","Text":"Both of these conditions hold,"},{"Start":"01:30.765 ","End":"01:35.600","Text":"so what is a conjugate and what do I mean by multiplying by the conjugate?"},{"Start":"01:35.600 ","End":"01:37.310","Text":"Well, I\u0027ll show you."},{"Start":"01:37.310 ","End":"01:39.290","Text":"If we have a difference of 2 terms,"},{"Start":"01:39.290 ","End":"01:40.730","Text":"A minus B,"},{"Start":"01:40.730 ","End":"01:45.320","Text":"then its conjugate is A plus B and vice versa."},{"Start":"01:45.320 ","End":"01:50.030","Text":"If we multiply an expression by its conjugate,"},{"Start":"01:50.030 ","End":"01:53.965","Text":"what we get is A minus B times A plus B."},{"Start":"01:53.965 ","End":"01:58.550","Text":"There\u0027s a formula that it\u0027s equal to A squared minus B squared."},{"Start":"01:58.550 ","End":"02:00.410","Text":"What is this good for?"},{"Start":"02:00.410 ","End":"02:05.675","Text":"Well, if A and/or B has a square root over it,"},{"Start":"02:05.675 ","End":"02:07.070","Text":"then when we square it,"},{"Start":"02:07.070 ","End":"02:09.380","Text":"we\u0027ll get rid of the square root."},{"Start":"02:09.380 ","End":"02:14.520","Text":"Let\u0027s see what happens in our example."},{"Start":"02:14.900 ","End":"02:22.915","Text":"I\u0027m taking the example of A and B from the numerator of our example here."},{"Start":"02:22.915 ","End":"02:25.000","Text":"Let\u0027s just, as a side exercise,"},{"Start":"02:25.000 ","End":"02:29.885","Text":"take that numerator and let\u0027s multiply it by its conjugate."},{"Start":"02:29.885 ","End":"02:32.050","Text":"Instead of the minus here,"},{"Start":"02:32.050 ","End":"02:33.515","Text":"I have a plus here,"},{"Start":"02:33.515 ","End":"02:38.180","Text":"and if we multiply them using this formula,"},{"Start":"02:38.700 ","End":"02:44.905","Text":"what we get is the square root"},{"Start":"02:44.905 ","End":"02:52.515","Text":"of 15 plus x squared,"},{"Start":"02:52.515 ","End":"02:58.695","Text":"that\u0027s my A squared minus B squared,"},{"Start":"02:58.695 ","End":"03:02.050","Text":"which is 4 squared."},{"Start":"03:02.300 ","End":"03:09.830","Text":"Now, taking the square of the square root just leaves us with the expression itself,"},{"Start":"03:09.830 ","End":"03:12.510","Text":"which is 15 plus x."},{"Start":"03:13.940 ","End":"03:19.490","Text":"Here, that\u0027s the essence of the multiplication with the conjugate,"},{"Start":"03:19.490 ","End":"03:22.610","Text":"is that if you take something can multiply by its conjugate,"},{"Start":"03:22.610 ","End":"03:24.790","Text":"you don\u0027t have any square roots anymore."},{"Start":"03:24.790 ","End":"03:28.170","Text":"It\u0027s 15 plus x minus this 4 squared,"},{"Start":"03:28.170 ","End":"03:36.180","Text":"which is 16, which ultimately just leaves us with x minus 1."},{"Start":"03:36.880 ","End":"03:39.080","Text":"Now, having done this,"},{"Start":"03:39.080 ","End":"03:41.405","Text":"we return to our example."},{"Start":"03:41.405 ","End":"03:45.020","Text":"How we would solve it is that we would take"},{"Start":"03:45.020 ","End":"03:51.910","Text":"this expression and write it in a slightly different form algebraically."},{"Start":"03:51.910 ","End":"03:55.780","Text":"X goes to 1 of,"},{"Start":"03:55.780 ","End":"03:58.655","Text":"first of all, I\u0027m going to copy the same expression,"},{"Start":"03:58.655 ","End":"04:06.440","Text":"15 plus x minus 4 over x minus 1."},{"Start":"04:06.440 ","End":"04:10.640","Text":"I would like to multiply this by its conjugate,"},{"Start":"04:10.640 ","End":"04:18.810","Text":"which is the square root of 15 plus x plus 4."},{"Start":"04:18.810 ","End":"04:21.920","Text":"But of course, you can\u0027t just multiply it by"},{"Start":"04:21.920 ","End":"04:24.619","Text":"something because it won\u0027t be the same exercise."},{"Start":"04:24.619 ","End":"04:32.975","Text":"I have to compensate and multiply the numerator or the denominator also by same thing,"},{"Start":"04:32.975 ","End":"04:38.810","Text":"square root of 15 plus x plus 4, same thing."},{"Start":"04:38.810 ","End":"04:42.800","Text":"Something over itself is just 1 and you can multiply by 1,"},{"Start":"04:42.800 ","End":"04:45.185","Text":"leave the expression unchanged."},{"Start":"04:45.185 ","End":"04:56.115","Text":"But if we do a little bit of algebra here and multiply it out,"},{"Start":"04:56.115 ","End":"04:59.060","Text":"well, this times this, we\u0027ve already done."},{"Start":"04:59.060 ","End":"05:00.860","Text":"In fact, we\u0027ve done it over here."},{"Start":"05:00.860 ","End":"05:03.035","Text":"No need to do it again."},{"Start":"05:03.035 ","End":"05:06.545","Text":"This equals the limit of"},{"Start":"05:06.545 ","End":"05:11.675","Text":"this times this is x minus 1 is all we have left on the numerator,"},{"Start":"05:11.675 ","End":"05:12.950","Text":"and on the denominator,"},{"Start":"05:12.950 ","End":"05:15.050","Text":"we have this thing times this thing,"},{"Start":"05:15.050 ","End":"05:20.290","Text":"so we have, just put it in brackets first."},{"Start":"05:20.290 ","End":"05:22.860","Text":"We have from here,"},{"Start":"05:22.860 ","End":"05:26.370","Text":"the x minus 1, and from here,"},{"Start":"05:26.370 ","End":"05:30.000","Text":"the square root of"},{"Start":"05:30.000 ","End":"05:39.240","Text":"15 plus x plus 4."},{"Start":"05:39.240 ","End":"05:43.955","Text":"Now, this is where we come to familiar territory."},{"Start":"05:43.955 ","End":"05:48.720","Text":"We can now use the factorizing canceling, in this case,"},{"Start":"05:48.720 ","End":"05:51.740","Text":"canceling technique, and just look x minus 1 and x"},{"Start":"05:51.740 ","End":"05:55.190","Text":"minus 1 because I have to leave a 1 here."},{"Start":"05:55.190 ","End":"05:57.620","Text":"It\u0027s going to be something on the numerator."},{"Start":"05:57.620 ","End":"06:00.245","Text":"Now in this expression,"},{"Start":"06:00.245 ","End":"06:02.570","Text":"there are no problems,"},{"Start":"06:02.570 ","End":"06:04.325","Text":"there\u0027s no zeros in the denominator."},{"Start":"06:04.325 ","End":"06:06.290","Text":"We could just use substitution."},{"Start":"06:06.290 ","End":"06:11.460","Text":"I\u0027m going to substitute x equals 1 in here,"},{"Start":"06:11.650 ","End":"06:20.670","Text":"and what we will get is 1 over,"},{"Start":"06:20.670 ","End":"06:22.635","Text":"let\u0027s see if we can do this in our heads,"},{"Start":"06:22.635 ","End":"06:26.145","Text":"15 plus 1 is 16,"},{"Start":"06:26.145 ","End":"06:30.780","Text":"square root of 16 is 4 plus 4 just comes out as 8,"},{"Start":"06:30.780 ","End":"06:35.370","Text":"so 1/8, that\u0027s the answer."},{"Start":"06:35.370 ","End":"06:37.905","Text":"Not too bad. A little bit of algebra."},{"Start":"06:37.905 ","End":"06:40.200","Text":"Be careful with those square roots."},{"Start":"06:40.200 ","End":"06:45.870","Text":"Mainly, to multiply by the conjugate and then we get rid of square roots."},{"Start":"06:49.510 ","End":"06:56.990","Text":"The essential step in this exercise I\u0027d like to point out was this product."},{"Start":"06:56.990 ","End":"06:59.765","Text":"We multiply by something over itself,"},{"Start":"06:59.765 ","End":"07:01.970","Text":"and 1 of these is the conjugate,"},{"Start":"07:01.970 ","End":"07:03.740","Text":"either the numerator or the denominator is"},{"Start":"07:03.740 ","End":"07:07.205","Text":"the conjugate of what we had in the original exercise."},{"Start":"07:07.205 ","End":"07:10.470","Text":"It\u0027s not too bad, but it\u0027s still fairly difficult,"},{"Start":"07:10.470 ","End":"07:14.075","Text":"you\u0027ll have plenty more such exercises to practice on,"},{"Start":"07:14.075 ","End":"07:17.675","Text":"but I would like to say something important,"},{"Start":"07:17.675 ","End":"07:20.280","Text":"that in the future,"},{"Start":"07:20.450 ","End":"07:23.190","Text":"these things will get a lot easier,"},{"Start":"07:23.190 ","End":"07:25.790","Text":"when you\u0027ve learned L\u0027Hopital\u0027s Rule,"},{"Start":"07:25.790 ","End":"07:27.665","Text":"I\u0027ll just write his name."},{"Start":"07:27.665 ","End":"07:29.280","Text":"He\u0027s a French,"},{"Start":"07:29.280 ","End":"07:31.670","Text":"I\u0027m not sure in which century."},{"Start":"07:31.670 ","End":"07:37.220","Text":"But there was a clever fellow called L\u0027Hopital,"},{"Start":"07:37.220 ","End":"07:41.820","Text":"and he had rules for limits,"},{"Start":"07:41.820 ","End":"07:46.070","Text":"mainly of the form 0/0 or infinity over infinity,"},{"Start":"07:46.070 ","End":"07:47.780","Text":"and once we\u0027ve learned these,"},{"Start":"07:47.780 ","End":"07:49.520","Text":"things will get a whole lot easier."},{"Start":"07:49.520 ","End":"07:55.640","Text":"Thing is that you have to know differentiation and the use of derivatives,"},{"Start":"07:55.640 ","End":"07:57.665","Text":"and once you\u0027ve learned those,"},{"Start":"07:57.665 ","End":"08:01.895","Text":"we can learn L\u0027Hopital\u0027s Rule, and then things will get a whole lot easier."},{"Start":"08:01.895 ","End":"08:06.095","Text":"Just to keep you encouraged or don\u0027t get discouraged."},{"Start":"08:06.095 ","End":"08:12.870","Text":"Anyway, that\u0027s it for multiplying by the conjugate. We\u0027re done."}],"ID":8410},{"Watched":false,"Name":"Exercise 1","Duration":"4m 3s","ChapterTopicVideoID":1532,"CourseChapterTopicPlaylistID":65361,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1532.jpeg","UploadDate":"2014-10-22T04:04:40.4630000","DurationForVideoObject":"PT4M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.250","Text":"In this exercise, we have to find the limit as x tends to 1 of the function,"},{"Start":"00:05.250 ","End":"00:08.955","Text":"1 minus the square root of x over 1 minus x."},{"Start":"00:08.955 ","End":"00:11.265","Text":"This is an elementary function."},{"Start":"00:11.265 ","End":"00:16.365","Text":"If we\u0027re lucky, we can just substitute x equals 1 and get the answer."},{"Start":"00:16.365 ","End":"00:19.950","Text":"Unfortunately, we\u0027re not because if you substitute x"},{"Start":"00:19.950 ","End":"00:23.525","Text":"equals 1 in the denominator, you get 0."},{"Start":"00:23.525 ","End":"00:25.860","Text":"In the numerator, square root of 1 is 1,"},{"Start":"00:25.860 ","End":"00:28.215","Text":"1 minus 1 is also 0."},{"Start":"00:28.215 ","End":"00:34.560","Text":"We have a limit of the form 0 over 0 and we have to apply some technique."},{"Start":"00:34.560 ","End":"00:39.780","Text":"Looking at it, we see that 1 of the terms has a square root over it."},{"Start":"00:39.780 ","End":"00:43.670","Text":"This indicates using the method of the conjugate."},{"Start":"00:43.670 ","End":"00:45.845","Text":"What is a conjugate in general?"},{"Start":"00:45.845 ","End":"00:49.460","Text":"In general, the conjugate of"},{"Start":"00:49.460 ","End":"00:56.309","Text":"a plus b is a minus b and vice versa."},{"Start":"00:56.309 ","End":"00:57.965","Text":"Each conjugates of the other."},{"Start":"00:57.965 ","End":"01:01.850","Text":"Interesting thing, or the useful thing about conjugates is that if you"},{"Start":"01:01.850 ","End":"01:07.330","Text":"multiply them a plus b times a minus b,"},{"Start":"01:07.330 ","End":"01:09.965","Text":"using the difference of squares formula,"},{"Start":"01:09.965 ","End":"01:12.950","Text":"we get a squared minus b squared."},{"Start":"01:12.950 ","End":"01:15.500","Text":"If a or b is a square root sign,"},{"Start":"01:15.500 ","End":"01:19.675","Text":"after expanding here, the square root sign will disappear."},{"Start":"01:19.675 ","End":"01:22.880","Text":"This would be useful for us in our case."},{"Start":"01:22.880 ","End":"01:25.970","Text":"Now, let\u0027s take a look at our numerator."},{"Start":"01:25.970 ","End":"01:30.935","Text":"If we take this as,1 as a and square root of x as b,"},{"Start":"01:30.935 ","End":"01:35.150","Text":"then the conjugate would be 1 plus the square root of x."},{"Start":"01:35.150 ","End":"01:38.255","Text":"If we multiply, we\u0027ll get rid of the square roots."},{"Start":"01:38.255 ","End":"01:41.410","Text":"Let\u0027s just play with the numerator at first."},{"Start":"01:41.410 ","End":"01:45.950","Text":"If we see what is 1 minus the square root of"},{"Start":"01:45.950 ","End":"01:51.410","Text":"x and multiply it by its conjugate 1 plus the square root of x,"},{"Start":"01:51.410 ","End":"01:56.480","Text":"we\u0027ll get a squared minus b squared is 1 squared,"},{"Start":"01:56.480 ","End":"02:01.520","Text":"which is 1 minus the square root of x squared is just x."},{"Start":"02:01.520 ","End":"02:04.610","Text":"This will help us to write this fraction in"},{"Start":"02:04.610 ","End":"02:08.360","Text":"a different form where it will be easier to find the limit."},{"Start":"02:08.360 ","End":"02:16.910","Text":"Let\u0027s see, the limit as x tends to 1 of 1 minus the square root of"},{"Start":"02:16.910 ","End":"02:25.460","Text":"x over 1 minus x will equal the limit as x goes to 1."},{"Start":"02:25.460 ","End":"02:31.490","Text":"Now what I want to do is multiply the numerator by the conjugate 1 plus square root of x."},{"Start":"02:31.490 ","End":"02:34.580","Text":"But I can\u0027t just multiply the numerator."},{"Start":"02:34.580 ","End":"02:39.080","Text":"What I\u0027d like to do then is to multiply this by 1"},{"Start":"02:39.080 ","End":"02:43.190","Text":"plus the square root of x and that will simplify it."},{"Start":"02:43.190 ","End":"02:46.609","Text":"But as I say, you can\u0027t just multiply the numerator of a fraction."},{"Start":"02:46.609 ","End":"02:49.370","Text":"If I multiply the denominator also,"},{"Start":"02:49.370 ","End":"02:55.610","Text":"then what I\u0027m really doing is I\u0027m multiplying the fraction by 1,"},{"Start":"02:55.610 ","End":"02:57.950","Text":"so I\u0027m not changing it in any way."},{"Start":"02:57.950 ","End":"03:04.520","Text":"If we continue, we get the limit x goes to 1 of."},{"Start":"03:04.520 ","End":"03:12.725","Text":"Now this times this we\u0027ve already done as a side exercise over here and that\u0027s 1 minus x."},{"Start":"03:12.725 ","End":"03:16.370","Text":"The denominator is this times this,"},{"Start":"03:16.370 ","End":"03:24.014","Text":"so it\u0027s 1 minus x times 1 plus the square root x."},{"Start":"03:24.014 ","End":"03:29.280","Text":"Here we are lucky that 1 minus x cancels."},{"Start":"03:29.280 ","End":"03:32.690","Text":"In the numerator, we\u0027re left with just 1."},{"Start":"03:32.690 ","End":"03:36.200","Text":"We can cancel because x tends to 1,"},{"Start":"03:36.200 ","End":"03:37.865","Text":"but it is not equal to 1."},{"Start":"03:37.865 ","End":"03:42.845","Text":"Now all we\u0027re left with is 1 over 1 plus the square root of x."},{"Start":"03:42.845 ","End":"03:50.390","Text":"This is also an elementary function and this time we can substitute x equals 1 here."},{"Start":"03:50.390 ","End":"03:58.475","Text":"We get that this equals 1 over 1 plus the square root of 1."},{"Start":"03:58.475 ","End":"04:04.530","Text":"Obviously this is equal to 1/2 and that\u0027s our answer."}],"ID":1544},{"Watched":false,"Name":"Exercise 2","Duration":"4m 29s","ChapterTopicVideoID":1533,"CourseChapterTopicPlaylistID":65361,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1533.jpeg","UploadDate":"2014-10-22T04:05:01.2470000","DurationForVideoObject":"PT4M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.500","Text":"In this exercise, we have to find the limit as x tends to 3"},{"Start":"00:04.500 ","End":"00:10.980","Text":"of this function of x minus 3 over the square root of x plus 1 minus 2."},{"Start":"00:10.980 ","End":"00:12.720","Text":"It\u0027s an elementary function,"},{"Start":"00:12.720 ","End":"00:14.280","Text":"and if we\u0027re lucky,"},{"Start":"00:14.280 ","End":"00:17.910","Text":"we could just substitute x equals 3 and get the answer."},{"Start":"00:17.910 ","End":"00:19.935","Text":"But this is not so."},{"Start":"00:19.935 ","End":"00:23.160","Text":"Because if we put x equals 3 in the numerator,"},{"Start":"00:23.160 ","End":"00:24.540","Text":"we get 3 minus 3,"},{"Start":"00:24.540 ","End":"00:26.370","Text":"which is 0,"},{"Start":"00:26.370 ","End":"00:31.025","Text":"and if we put x equals 3 in the denominator, is also 0."},{"Start":"00:31.025 ","End":"00:34.190","Text":"So we have an expression of the form 0 over 0,"},{"Start":"00:34.190 ","End":"00:36.905","Text":"and we have to use some technique."},{"Start":"00:36.905 ","End":"00:40.655","Text":"I notice that there is a square root here,"},{"Start":"00:40.655 ","End":"00:43.550","Text":"and if 1 of the terms or more in"},{"Start":"00:43.550 ","End":"00:46.640","Text":"the numerator or denominator is under a square root sign,"},{"Start":"00:46.640 ","End":"00:50.330","Text":"that indicates that we should probably be using conjugates."},{"Start":"00:50.330 ","End":"00:52.115","Text":"What is a conjugate?"},{"Start":"00:52.115 ","End":"00:57.680","Text":"Well, in general, the conjugate of an expression A plus B"},{"Start":"00:57.680 ","End":"01:03.780","Text":"is just simply the opposite sign A minus B,"},{"Start":"01:03.780 ","End":"01:08.810","Text":"and conversely, the conjugate of A minus B is A plus B."},{"Start":"01:08.810 ","End":"01:13.885","Text":"This is often useful because when you multiply 2 conjugates,"},{"Start":"01:13.885 ","End":"01:18.375","Text":"in this case, A plus B times A minus B,"},{"Start":"01:18.375 ","End":"01:25.775","Text":"we get A squared minus B squared using the famous difference of squares formula."},{"Start":"01:25.775 ","End":"01:28.760","Text":"So if A or B was a square root,"},{"Start":"01:28.760 ","End":"01:30.470","Text":"after we square that,"},{"Start":"01:30.470 ","End":"01:32.150","Text":"it will no longer be a square root,"},{"Start":"01:32.150 ","End":"01:34.735","Text":"and that usually helps."},{"Start":"01:34.735 ","End":"01:38.225","Text":"In our case, here\u0027s the square root."},{"Start":"01:38.225 ","End":"01:42.155","Text":"I proposed to let the square root of x plus 1 be A,"},{"Start":"01:42.155 ","End":"01:45.500","Text":"and 2 can be B, and we\u0027ll have A minus B."},{"Start":"01:45.500 ","End":"01:47.539","Text":"Let\u0027s first of all, as an exercise,"},{"Start":"01:47.539 ","End":"01:51.520","Text":"see what happens if we multiply the denominator by its conjugate?"},{"Start":"01:51.520 ","End":"01:57.080","Text":"A square root of x plus 1 minus 2,"},{"Start":"01:57.080 ","End":"01:59.225","Text":"and I\u0027ll use brackets here."},{"Start":"01:59.225 ","End":"02:01.580","Text":"I\u0027ll multiply it by its conjugate,"},{"Start":"02:01.580 ","End":"02:06.305","Text":"which is the square root of x plus 1 plus 2."},{"Start":"02:06.305 ","End":"02:10.000","Text":"This equals, using this formula,"},{"Start":"02:10.000 ","End":"02:12.170","Text":"A squared minus B squared."},{"Start":"02:12.170 ","End":"02:19.160","Text":"The square root of x plus 1 squared is just x plus 1 without the square root,"},{"Start":"02:19.160 ","End":"02:22.685","Text":"and then minus the 2 squared."},{"Start":"02:22.685 ","End":"02:28.160","Text":"In short, this is just equal to x minus 3."},{"Start":"02:28.160 ","End":"02:31.280","Text":"This is going to help us to solve this limit."},{"Start":"02:31.280 ","End":"02:32.855","Text":"Let\u0027s continue here."},{"Start":"02:32.855 ","End":"02:40.640","Text":"The limit as x tends to 3 of x minus 3"},{"Start":"02:40.640 ","End":"02:47.135","Text":"over square root of x plus 1 minus 2."},{"Start":"02:47.135 ","End":"02:55.850","Text":"What we can do is to multiply top and bottom by the conjugate of the denominator."},{"Start":"02:55.850 ","End":"02:58.895","Text":"I\u0027d like to multiply this by its conjugate,"},{"Start":"02:58.895 ","End":"03:03.990","Text":"which is square root of x plus 1 plus 2."},{"Start":"03:03.990 ","End":"03:06.105","Text":"But I can\u0027t just multiply it,"},{"Start":"03:06.105 ","End":"03:09.680","Text":"I have to compensate by multiplying the numerator also."},{"Start":"03:09.680 ","End":"03:14.375","Text":"Here again, I\u0027m going to write the square root of x plus 1 plus 2."},{"Start":"03:14.375 ","End":"03:18.650","Text":"In fact, this whole thing is just equal to 1."},{"Start":"03:18.650 ","End":"03:22.235","Text":"So I can multiply, haven\u0027t changed anything."},{"Start":"03:22.235 ","End":"03:29.450","Text":"Now, this equals the limit as x tends to 3."},{"Start":"03:29.450 ","End":"03:32.420","Text":"I\u0027ll do the denominator first"},{"Start":"03:32.420 ","End":"03:33.995","Text":"because we\u0027ve actually done that."},{"Start":"03:33.995 ","End":"03:36.245","Text":"This times this, we\u0027ve already done up here,"},{"Start":"03:36.245 ","End":"03:38.195","Text":"and we get x minus 3."},{"Start":"03:38.195 ","End":"03:40.895","Text":"The numerator, this bit times this bit,"},{"Start":"03:40.895 ","End":"03:46.505","Text":"I\u0027ve got square root of x plus 1 plus 2,"},{"Start":"03:46.505 ","End":"03:50.285","Text":"and look we have x minus 3 here and x minus 3 here."},{"Start":"03:50.285 ","End":"03:53.120","Text":"We can cancel this with this."},{"Start":"03:53.120 ","End":"03:55.819","Text":"We haven\u0027t canceled by 0,"},{"Start":"03:55.819 ","End":"03:59.180","Text":"because the x tends to 3 but doesn\u0027t equal 3,"},{"Start":"03:59.180 ","End":"04:01.190","Text":"and that\u0027s why this is not 0."},{"Start":"04:01.190 ","End":"04:04.610","Text":"All we\u0027re left with is this bit in the brackets,"},{"Start":"04:04.610 ","End":"04:06.980","Text":"square root of x plus 1 plus 2."},{"Start":"04:06.980 ","End":"04:09.660","Text":"That\u0027s certainly elementary function,"},{"Start":"04:09.660 ","End":"04:12.345","Text":"and this time we can substitute 3."},{"Start":"04:12.345 ","End":"04:17.750","Text":"This is going to equal just substituting 3 for x,"},{"Start":"04:17.750 ","End":"04:24.260","Text":"we get the square root of 3 plus 1 plus 2,"},{"Start":"04:24.260 ","End":"04:26.435","Text":"which is 4,"},{"Start":"04:26.435 ","End":"04:29.550","Text":"and that\u0027s the answer."}],"ID":1545},{"Watched":false,"Name":"Exercise 3","Duration":"5m 22s","ChapterTopicVideoID":1534,"CourseChapterTopicPlaylistID":65361,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1534.jpeg","UploadDate":"2014-10-22T04:05:26.9170000","DurationForVideoObject":"PT5M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.705","Text":"In this exercise, we want to find the limit as x tends to 3 of this function,"},{"Start":"00:06.705 ","End":"00:12.120","Text":"3 minus the square root of x plus 6 over 2x minus 6."},{"Start":"00:12.120 ","End":"00:15.000","Text":"It\u0027s elementary as a function."},{"Start":"00:15.000 ","End":"00:19.425","Text":"Hopefully, we could just substitute x equals 3."},{"Start":"00:19.425 ","End":"00:25.410","Text":"Unfortunately, that doesn\u0027t work because if we put x equals 3 in the denominator,"},{"Start":"00:25.410 ","End":"00:26.460","Text":"we get 0,"},{"Start":"00:26.460 ","End":"00:28.935","Text":"twice 3 minus 6 is 0."},{"Start":"00:28.935 ","End":"00:31.590","Text":"In the numerator,"},{"Start":"00:31.590 ","End":"00:34.935","Text":"we\u0027re in a 0 over 0 situation."},{"Start":"00:34.935 ","End":"00:40.970","Text":"We have to see what tools we could use to help us do this, simplify it."},{"Start":"00:40.970 ","End":"00:45.665","Text":"I notice that there\u0027s a square root in 1 of the terms in the numerator."},{"Start":"00:45.665 ","End":"00:48.485","Text":"Whenever there\u0027s a square root or more"},{"Start":"00:48.485 ","End":"00:51.590","Text":"in one of the terms in the numerator or denominator,"},{"Start":"00:51.590 ","End":"00:55.505","Text":"it usually indicates that we should try the method of conjugates."},{"Start":"00:55.505 ","End":"00:58.250","Text":"I\u0027ll remind you what a conjugate is,"},{"Start":"00:58.250 ","End":"01:01.220","Text":"in general, of a sum or difference, let\u0027s say,"},{"Start":"01:01.220 ","End":"01:05.000","Text":"A plus B is just the same thing with the opposite sign,"},{"Start":"01:05.000 ","End":"01:07.085","Text":"if it\u0027s a plus, it\u0027s a minus,"},{"Start":"01:07.085 ","End":"01:09.925","Text":"is A minus B."},{"Start":"01:09.925 ","End":"01:12.840","Text":"In other words, each is a conjugate of the other."},{"Start":"01:12.840 ","End":"01:16.610","Text":"The thing about conjugates that makes them useful is that"},{"Start":"01:16.610 ","End":"01:24.370","Text":"if you multiply a conjugate pair, A plus B times A minus B,"},{"Start":"01:24.370 ","End":"01:30.905","Text":"we get, using the difference of squares formula, A squared minus B squared."},{"Start":"01:30.905 ","End":"01:33.770","Text":"If either A or B is a square root,"},{"Start":"01:33.770 ","End":"01:36.080","Text":"when we square it, we get rid of the square root,"},{"Start":"01:36.080 ","End":"01:38.220","Text":"so this could be useful."},{"Start":"01:38.220 ","End":"01:41.570","Text":"Let\u0027s see what happens in our case."},{"Start":"01:41.570 ","End":"01:44.375","Text":"The numerator is the one with the square root."},{"Start":"01:44.375 ","End":"01:49.445","Text":"What we\u0027d like to do is to multiply the numerator by its conjugate."},{"Start":"01:49.445 ","End":"01:53.760","Text":"Let\u0027s just do that as a side exercise first."},{"Start":"01:53.760 ","End":"01:55.320","Text":"This is A minus B,"},{"Start":"01:55.320 ","End":"01:57.090","Text":"we multiply by A plus B,"},{"Start":"01:57.090 ","End":"02:06.185","Text":"we get 3 minus the square root of x plus 6 times its conjugate,"},{"Start":"02:06.185 ","End":"02:09.425","Text":"which is if it\u0027s a minus, it\u0027s a plus, and vice versa,"},{"Start":"02:09.425 ","End":"02:13.845","Text":"3 plus the square root of x plus 6."},{"Start":"02:13.845 ","End":"02:14.940","Text":"This is equal to,"},{"Start":"02:14.940 ","End":"02:18.065","Text":"using this formula of A squared minus B squared,"},{"Start":"02:18.065 ","End":"02:25.740","Text":"A squared is 3 squared is 9 minus, the other part squared,"},{"Start":"02:25.740 ","End":"02:30.860","Text":"so the square root comes off and we\u0027re just left with, x plus 6."},{"Start":"02:30.860 ","End":"02:36.280","Text":"This simplifies to 3 minus x."},{"Start":"02:36.280 ","End":"02:41.030","Text":"Now, let\u0027s try using this to help us to solve this limit,"},{"Start":"02:41.030 ","End":"02:42.860","Text":"so I\u0027ll rewrite it."},{"Start":"02:42.860 ","End":"02:45.350","Text":"What I really want to do is to multiply"},{"Start":"02:45.350 ","End":"02:49.700","Text":"this 3 minus the square root by 3 plus the square root by its conjugate."},{"Start":"02:49.700 ","End":"02:55.950","Text":"I\u0027m going to multiply by 3 plus the square root of x plus 6."},{"Start":"02:55.950 ","End":"03:00.765","Text":"But I can\u0027t just multiply because I\u0027ve changed the exercise."},{"Start":"03:00.765 ","End":"03:06.910","Text":"But if I multiply the denominator also by the same thing,"},{"Start":"03:06.910 ","End":"03:11.610","Text":"this whole thing is just equal to 1,"},{"Start":"03:11.610 ","End":"03:18.435","Text":"and I can always multiply by 1 without changing anything, so that\u0027s allowed."},{"Start":"03:18.435 ","End":"03:23.290","Text":"What we get is limit x tends to 3."},{"Start":"03:23.290 ","End":"03:27.300","Text":"Start with the numerator because we\u0027ve already done this."},{"Start":"03:27.300 ","End":"03:31.760","Text":"This numerator times this numerator is exactly what we did here,"},{"Start":"03:31.760 ","End":"03:35.509","Text":"so we can write the answer as 3 minus x."},{"Start":"03:35.509 ","End":"03:38.060","Text":"Then in the denominator,"},{"Start":"03:38.060 ","End":"03:41.410","Text":"we just have to multiply this by this,"},{"Start":"03:41.410 ","End":"03:47.235","Text":"2x minus 6 times"},{"Start":"03:47.235 ","End":"03:53.595","Text":"3 plus the square root of x plus 6."},{"Start":"03:53.595 ","End":"03:56.100","Text":"Now, previously in exercises of this sort,"},{"Start":"03:56.100 ","End":"03:58.005","Text":"there was a factor that canceled,"},{"Start":"03:58.005 ","End":"04:03.170","Text":"and here, it doesn\u0027t seem to be a factor that appears both in numerator and denominator."},{"Start":"04:03.170 ","End":"04:05.000","Text":"But if you look more closely,"},{"Start":"04:05.000 ","End":"04:08.525","Text":"2x minus 6 is not very different from 3 minus x."},{"Start":"04:08.525 ","End":"04:10.235","Text":"If we reverse the order,"},{"Start":"04:10.235 ","End":"04:12.190","Text":"it was x minus 3,"},{"Start":"04:12.190 ","End":"04:14.265","Text":"this is exactly double this."},{"Start":"04:14.265 ","End":"04:20.810","Text":"I\u0027d like to make a note at the side that 2x minus 6 is"},{"Start":"04:20.810 ","End":"04:28.620","Text":"simply equal to minus twice 3 minus x."},{"Start":"04:28.620 ","End":"04:30.815","Text":"If I notice this,"},{"Start":"04:30.815 ","End":"04:33.855","Text":"then I can do a cancellation of sorts."},{"Start":"04:33.855 ","End":"04:37.759","Text":"I can cancel this with this,"},{"Start":"04:37.759 ","End":"04:42.295","Text":"but I have to leave a minus 2 here."},{"Start":"04:42.295 ","End":"04:48.090","Text":"Continuing, this a 1 in the numerator,"},{"Start":"04:48.090 ","End":"04:52.855","Text":"all I have is 1 over minus twice 3 plus the square root."},{"Start":"04:52.855 ","End":"04:55.190","Text":"This is also an elementary function,"},{"Start":"04:55.190 ","End":"04:58.850","Text":"but this time we can substitute x equals 3."},{"Start":"04:58.850 ","End":"05:00.455","Text":"If we do that,"},{"Start":"05:00.455 ","End":"05:05.765","Text":"we\u0027ll get the answer which is 1 over"},{"Start":"05:05.765 ","End":"05:15.035","Text":"minus 2 times 3 plus square root of 3 plus 6,"},{"Start":"05:15.035 ","End":"05:20.790","Text":"so the answer is just minus 1 over 12."},{"Start":"05:20.790 ","End":"05:23.890","Text":"That\u0027s the answer. We\u0027re done."}],"ID":1546},{"Watched":false,"Name":"Exercise 4","Duration":"8m 28s","ChapterTopicVideoID":1535,"CourseChapterTopicPlaylistID":65361,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1535.jpeg","UploadDate":"2014-10-22T04:06:09.5770000","DurationForVideoObject":"PT8M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.135","Text":"In this exercise, we\u0027re asked to find the limit as x tends to 1 of the function,"},{"Start":"00:06.135 ","End":"00:10.515","Text":"square root of x squared plus x plus 2 minus 2,"},{"Start":"00:10.515 ","End":"00:13.095","Text":"all over x squared minus 1."},{"Start":"00:13.095 ","End":"00:15.255","Text":"This is an elementary function,"},{"Start":"00:15.255 ","End":"00:19.830","Text":"we would like to hope that we could just substitute x equals 1."},{"Start":"00:19.830 ","End":"00:24.285","Text":"Unfortunately, that doesn\u0027t work because if we do substitute x equals 1,"},{"Start":"00:24.285 ","End":"00:26.400","Text":"we get 1 squared minus 1,"},{"Start":"00:26.400 ","End":"00:30.915","Text":"which is 0, and in the numerator is 0."},{"Start":"00:30.915 ","End":"00:33.660","Text":"We\u0027re in the 0 over 0 situation"},{"Start":"00:33.660 ","End":"00:36.195","Text":"and we have to see what tools we can use."},{"Start":"00:36.195 ","End":"00:39.480","Text":"Well, the biggest hint is that there is a square root here,"},{"Start":"00:39.480 ","End":"00:41.960","Text":"and there\u0027s a couple of terms in the numerator,"},{"Start":"00:41.960 ","End":"00:43.910","Text":"a couple of terms in the denominator,"},{"Start":"00:43.910 ","End":"00:46.070","Text":"and the square root."},{"Start":"00:46.070 ","End":"00:49.810","Text":"This usually indicates the use of the conjugate,"},{"Start":"00:49.810 ","End":"00:51.200","Text":"and in case you forgotten,"},{"Start":"00:51.200 ","End":"00:53.465","Text":"I\u0027ll remind you what a conjugate is."},{"Start":"00:53.465 ","End":"00:58.895","Text":"In general, if you have a term of the form A plus B,"},{"Start":"00:58.895 ","End":"01:01.205","Text":"then I\u0027ll put it in brackets,"},{"Start":"01:01.205 ","End":"01:03.635","Text":"it has a conjugate term,"},{"Start":"01:03.635 ","End":"01:06.080","Text":"which is A minus B."},{"Start":"01:06.080 ","End":"01:09.650","Text":"Just the reverse sign works both ways."},{"Start":"01:09.650 ","End":"01:12.695","Text":"These 2 are conjugates of each other."},{"Start":"01:12.695 ","End":"01:16.930","Text":"This is useful because if you multiply 2 conjugates,"},{"Start":"01:16.930 ","End":"01:21.960","Text":"in this case A plus B by A minus B,"},{"Start":"01:21.960 ","End":"01:27.230","Text":"you get A squared minus B squared using the difference of squares formula,"},{"Start":"01:27.230 ","End":"01:31.085","Text":"and if A or B happens to be a square root,"},{"Start":"01:31.085 ","End":"01:33.830","Text":"which it will be, then when you square it,"},{"Start":"01:33.830 ","End":"01:35.975","Text":"the square root will just drop off."},{"Start":"01:35.975 ","End":"01:37.370","Text":"That will be helpful."},{"Start":"01:37.370 ","End":"01:40.285","Text":"Let\u0027s see how it helps us in our case."},{"Start":"01:40.285 ","End":"01:42.705","Text":"The numerator is the 1 with the square root,"},{"Start":"01:42.705 ","End":"01:45.950","Text":"so let\u0027s try and see what happens if we"},{"Start":"01:45.950 ","End":"01:49.310","Text":"look at the conjugate of the numerator and then multiply."},{"Start":"01:49.310 ","End":"01:52.160","Text":"We\u0027ll do this as a side exercise at first,"},{"Start":"01:52.160 ","End":"02:00.600","Text":"so take the numerator of x squared plus x plus 2 minus 2."},{"Start":"02:00.600 ","End":"02:03.375","Text":"That\u0027s like the A minus B,"},{"Start":"02:03.375 ","End":"02:07.490","Text":"and its conjugate will be the same thing with a plus,"},{"Start":"02:07.490 ","End":"02:15.754","Text":"so square root of x squared plus x plus 2, this time plus 2."},{"Start":"02:15.754 ","End":"02:18.215","Text":"If we multiply this out,"},{"Start":"02:18.215 ","End":"02:20.550","Text":"we get the A squared,"},{"Start":"02:20.550 ","End":"02:21.770","Text":"which is this 1 squared,"},{"Start":"02:21.770 ","End":"02:25.265","Text":"which is x squared plus x plus 2,"},{"Start":"02:25.265 ","End":"02:28.700","Text":"because a square root drops off, minus 2 squared."},{"Start":"02:28.700 ","End":"02:30.095","Text":"I\u0027ll just write it like that."},{"Start":"02:30.095 ","End":"02:31.340","Text":"That\u0027s obviously 4,"},{"Start":"02:31.340 ","End":"02:37.320","Text":"but we end up getting x squared plus x minus 2,"},{"Start":"02:37.320 ","End":"02:40.720","Text":"because it is 2 minus 4 is minus 2."},{"Start":"02:40.910 ","End":"02:44.030","Text":"That\u0027s a side exercise."},{"Start":"02:44.030 ","End":"02:46.445","Text":"Now see how it helps us solve our limit."},{"Start":"02:46.445 ","End":"02:48.215","Text":"Let\u0027s look again the limit."},{"Start":"02:48.215 ","End":"02:54.270","Text":"The limit as x tends to 1 of,"},{"Start":"02:54.270 ","End":"02:55.880","Text":"just copy it out again."},{"Start":"02:55.880 ","End":"02:57.289","Text":"To make things simple,"},{"Start":"02:57.289 ","End":"03:02.825","Text":"what I want to do is to multiply this thing by the conjugate,"},{"Start":"03:02.825 ","End":"03:04.130","Text":"like we did above,"},{"Start":"03:04.130 ","End":"03:10.300","Text":"which is the square root of x squared plus x plus 2 plus 2."},{"Start":"03:10.300 ","End":"03:13.740","Text":"But I can\u0027t just multiply the numerator,"},{"Start":"03:13.740 ","End":"03:15.500","Text":"that changes the exercise."},{"Start":"03:15.500 ","End":"03:21.050","Text":"However, if I also multiply the denominator by the same thing,"},{"Start":"03:21.050 ","End":"03:23.225","Text":"just copy it out again."},{"Start":"03:23.225 ","End":"03:28.235","Text":"Then essentially I multiply this fraction by this whole thing,"},{"Start":"03:28.235 ","End":"03:30.335","Text":"which happens to equal 1,"},{"Start":"03:30.335 ","End":"03:32.540","Text":"so I haven\u0027t changed anything."},{"Start":"03:32.540 ","End":"03:35.350","Text":"Let\u0027s continue and see what this is equal to."},{"Start":"03:35.350 ","End":"03:39.915","Text":"This is equal to the limit as x goes to 1."},{"Start":"03:39.915 ","End":"03:41.690","Text":"This times this we\u0027ve already done,"},{"Start":"03:41.690 ","End":"03:43.040","Text":"that\u0027s what we just did here,"},{"Start":"03:43.040 ","End":"03:50.600","Text":"so we can write the final answer for the numerator is x squared plus x minus 2,"},{"Start":"03:50.600 ","End":"03:52.125","Text":"and on the denominator,"},{"Start":"03:52.125 ","End":"03:54.165","Text":"something a bit more complicated,"},{"Start":"03:54.165 ","End":"04:00.664","Text":"x squared minus 1 times, in brackets,"},{"Start":"04:00.664 ","End":"04:09.720","Text":"the square root of x squared plus x plus 2 plus 2."},{"Start":"04:09.720 ","End":"04:12.875","Text":"Now, hopefully, we\u0027d like to cancel something,"},{"Start":"04:12.875 ","End":"04:16.665","Text":"but there\u0027s no obvious factor to cancel,"},{"Start":"04:16.665 ","End":"04:21.290","Text":"and usually, the idea is to factor quadratic expressions,"},{"Start":"04:21.290 ","End":"04:22.580","Text":"or whatever we can factor,"},{"Start":"04:22.580 ","End":"04:24.980","Text":"and hopefully, these things will have a common factor."},{"Start":"04:24.980 ","End":"04:28.590","Text":"Now, we know how to factor quadratic expressions."},{"Start":"04:28.590 ","End":"04:29.750","Text":"Here we have a quadratic,"},{"Start":"04:29.750 ","End":"04:31.205","Text":"here we have a quadratic."},{"Start":"04:31.205 ","End":"04:37.310","Text":"Let\u0027s take time off to just factorize these and then return to the main exercise."},{"Start":"04:37.310 ","End":"04:40.055","Text":"I want to remind you of another formula,"},{"Start":"04:40.055 ","End":"04:44.030","Text":"and I\u0027ll write it at the side, and that is,"},{"Start":"04:44.030 ","End":"04:51.575","Text":"that if we have an expression such as x squared plus bx plus c,"},{"Start":"04:51.575 ","End":"04:56.710","Text":"then this factors to x minus"},{"Start":"04:56.710 ","End":"05:02.220","Text":"x_1 times x minus x_2,"},{"Start":"05:02.220 ","End":"05:10.785","Text":"where x_1 and x_2 are the roots of the quadratic equation."},{"Start":"05:10.785 ","End":"05:14.150","Text":"Let\u0027s write it the roots of the equation by which"},{"Start":"05:14.150 ","End":"05:17.660","Text":"I mean x squared plus bx plus c equals 0."},{"Start":"05:17.660 ","End":"05:19.085","Text":"There as we solved this,"},{"Start":"05:19.085 ","End":"05:21.890","Text":"get the 2 roots, and then we can factor it this way."},{"Start":"05:21.890 ","End":"05:24.575","Text":"Let\u0027s see what happens in our case."},{"Start":"05:24.575 ","End":"05:26.395","Text":"We\u0027ll use this twice,"},{"Start":"05:26.395 ","End":"05:32.495","Text":"once on the numerator and also on the denominator."},{"Start":"05:32.495 ","End":"05:35.675","Text":"Let\u0027s do the green 1 first."},{"Start":"05:35.675 ","End":"05:38.870","Text":"First of all, we solve the equation,"},{"Start":"05:38.870 ","End":"05:43.715","Text":"x squared minus 1 equals 0."},{"Start":"05:43.715 ","End":"05:44.900","Text":"Well, in this case,"},{"Start":"05:44.900 ","End":"05:47.390","Text":"we didn\u0027t really need to do it this way,"},{"Start":"05:47.390 ","End":"05:49.280","Text":"there\u0027s actually a shorter way."},{"Start":"05:49.280 ","End":"05:53.210","Text":"We could have solved it and got that x is plus or minus 1,"},{"Start":"05:53.210 ","End":"05:55.325","Text":"but the easiest thing maybe,"},{"Start":"05:55.325 ","End":"05:58.160","Text":"was just to notice that this is also a difference of squares,"},{"Start":"05:58.160 ","End":"06:00.475","Text":"x squared minus 1 squared."},{"Start":"06:00.475 ","End":"06:09.140","Text":"X squared minus 1 is going to equal x minus 1 times x plus 1."},{"Start":"06:09.140 ","End":"06:14.320","Text":"The more slightly trickier 1 is the blue 1."},{"Start":"06:14.320 ","End":"06:22.300","Text":"To do that, we have to solve x squared plus x minus 2 equals 0."},{"Start":"06:22.300 ","End":"06:25.465","Text":"I assume you know how to solve quadratic equations."},{"Start":"06:25.465 ","End":"06:26.860","Text":"I won\u0027t solve it for you,"},{"Start":"06:26.860 ","End":"06:28.930","Text":"I\u0027ll just tell you that the roots are,"},{"Start":"06:28.930 ","End":"06:30.835","Text":"those are the 2 solutions."},{"Start":"06:30.835 ","End":"06:36.940","Text":"Because of this, it means that just as we go from here to here,"},{"Start":"06:36.940 ","End":"06:45.305","Text":"this means that x squared plus x minus 2 is equal to x minus x_1 minus x_2,"},{"Start":"06:45.305 ","End":"06:47.755","Text":"so it\u0027s x minus 1."},{"Start":"06:47.755 ","End":"06:53.445","Text":"X minus minus 2 makes it x plus 2."},{"Start":"06:53.445 ","End":"06:56.030","Text":"We\u0027ve got for the denominator,"},{"Start":"06:56.030 ","End":"06:59.360","Text":"this factorization, and for the numerator,"},{"Start":"06:59.360 ","End":"07:02.180","Text":"we\u0027re going to use this factorization,"},{"Start":"07:02.180 ","End":"07:07.580","Text":"and let\u0027s go back to writing what the limit equals."},{"Start":"07:07.580 ","End":"07:10.355","Text":"I\u0027ll just put an arrow here,"},{"Start":"07:10.355 ","End":"07:16.100","Text":"and this is going to equal the limit x goes to 1."},{"Start":"07:16.100 ","End":"07:18.635","Text":"Here, I\u0027ve factorized it,"},{"Start":"07:18.635 ","End":"07:20.750","Text":"the blue 1, x minus 1,"},{"Start":"07:20.750 ","End":"07:23.165","Text":"x plus 2 over."},{"Start":"07:23.165 ","End":"07:25.070","Text":"Now the green 1,"},{"Start":"07:25.070 ","End":"07:27.515","Text":"x minus 1, x plus 1,"},{"Start":"07:27.515 ","End":"07:29.240","Text":"and this bit here,"},{"Start":"07:29.240 ","End":"07:32.015","Text":"which we have to also maintain,"},{"Start":"07:32.015 ","End":"07:35.134","Text":"now we can definitely see something to cancel."},{"Start":"07:35.134 ","End":"07:39.195","Text":"The x minus 1 goes with the x minus 1."},{"Start":"07:39.195 ","End":"07:44.180","Text":"Notice that x minus 1 is not 0 because x is not equal to 1,"},{"Start":"07:44.180 ","End":"07:45.500","Text":"it tends to 1"},{"Start":"07:45.500 ","End":"07:46.985","Text":"but it\u0027s not equal to 1."},{"Start":"07:46.985 ","End":"07:51.815","Text":"What we\u0027re left with is the limit as x goes to 1 of this function here,"},{"Start":"07:51.815 ","End":"07:56.210","Text":"and here there is no problem in substituting x equals 1,"},{"Start":"07:56.210 ","End":"08:01.835","Text":"so this time it is equal to just substituting x equals 1,"},{"Start":"08:01.835 ","End":"08:04.000","Text":"so let\u0027s see, I\u0027ll just write it underneath here."},{"Start":"08:04.000 ","End":"08:10.560","Text":"We get 1 plus 2 over 1 plus 1 times"},{"Start":"08:10.560 ","End":"08:18.420","Text":"the square root of 1 plus 1 plus 2 plus 2,"},{"Start":"08:18.420 ","End":"08:24.150","Text":"and let\u0027s see what this gives us. 1 plus 2 is 3,"},{"Start":"08:24.150 ","End":"08:29.920","Text":"so the answer is 3/8, and we\u0027re done."}],"ID":1547},{"Watched":false,"Name":"Exercise 5","Duration":"6m 9s","ChapterTopicVideoID":1536,"CourseChapterTopicPlaylistID":65361,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1536.jpeg","UploadDate":"2014-10-22T04:06:40.2800000","DurationForVideoObject":"PT6M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.245","Text":"In this exercise, we have to find the limit as x tends to 4 of the following function,"},{"Start":"00:07.245 ","End":"00:10.935","Text":"square root of 2x plus 1 minus the square root of x plus 5,"},{"Start":"00:10.935 ","End":"00:13.320","Text":"all over x minus 4."},{"Start":"00:13.320 ","End":"00:17.145","Text":"If we were able to substitute x equals 4 here,"},{"Start":"00:17.145 ","End":"00:19.860","Text":"that would be very nice and we\u0027d be done."},{"Start":"00:19.860 ","End":"00:22.755","Text":"But unfortunately, this is not the case."},{"Start":"00:22.755 ","End":"00:26.505","Text":"If we put x equals 4 in the denominator,"},{"Start":"00:26.505 ","End":"00:30.840","Text":"we get 4 minus 4, which is 0."},{"Start":"00:30.840 ","End":"00:33.870","Text":"If we try putting x equals 4 in the numerator, let\u0027s see,"},{"Start":"00:33.870 ","End":"00:36.435","Text":"twice 4 plus 1 is 9,"},{"Start":"00:36.435 ","End":"00:39.600","Text":"so we have square root of 9,"},{"Start":"00:39.600 ","End":"00:42.460","Text":"and 4 plus 5 is 9."},{"Start":"00:42.460 ","End":"00:46.595","Text":"We have also square root of 9 and it\u0027s a minus,"},{"Start":"00:46.595 ","End":"00:48.850","Text":"so it also gives us 0."},{"Start":"00:48.850 ","End":"00:52.680","Text":"Basically, we\u0027re in a 0 over 0 situation,"},{"Start":"00:52.680 ","End":"00:56.930","Text":"so we have to think of what techniques we know."},{"Start":"00:56.930 ","End":"00:59.330","Text":"The biggest hint is the square roots."},{"Start":"00:59.330 ","End":"01:03.365","Text":"When you have a term and 1 or more of them has a square root,"},{"Start":"01:03.365 ","End":"01:07.920","Text":"usually, it indicates the use of the conjugate."},{"Start":"01:07.930 ","End":"01:11.795","Text":"I want to remind you what a conjugate is."},{"Start":"01:11.795 ","End":"01:13.490","Text":"When you have a term,"},{"Start":"01:13.490 ","End":"01:19.250","Text":"an expression of the form A minus B or A plus B,"},{"Start":"01:19.250 ","End":"01:27.050","Text":"its conjugate is the 1 with the opposite sign and the conjugate of A minus B is A plus B."},{"Start":"01:27.050 ","End":"01:29.230","Text":"Each of these is the conjugate of the other,"},{"Start":"01:29.230 ","End":"01:32.840","Text":"so these are conjugates."},{"Start":"01:32.840 ","End":"01:35.225","Text":"Now, the useful thing about conjugates,"},{"Start":"01:35.225 ","End":"01:37.490","Text":"that if you multiply a pair of them,"},{"Start":"01:37.490 ","End":"01:39.460","Text":"let\u0027s see what we get."},{"Start":"01:39.460 ","End":"01:43.040","Text":"A minus B times"},{"Start":"01:43.040 ","End":"01:50.315","Text":"A plus B is equal to A squared minus B squared."},{"Start":"01:50.315 ","End":"01:54.230","Text":"There\u0027s a difference of squares formula which gives us this."},{"Start":"01:54.230 ","End":"01:58.595","Text":"Now, if A or B or both were square roots,"},{"Start":"01:58.595 ","End":"01:59.960","Text":"when we square them,"},{"Start":"01:59.960 ","End":"02:01.240","Text":"we get rid of the square root,"},{"Start":"02:01.240 ","End":"02:03.710","Text":"so that\u0027s why conjugates are useful."},{"Start":"02:03.710 ","End":"02:06.470","Text":"Let\u0027s see how they can help us in our case."},{"Start":"02:06.470 ","End":"02:11.720","Text":"Here, the numerator is the 1 that has square roots, so its numerator,"},{"Start":"02:11.720 ","End":"02:14.389","Text":"that\u0027s interesting and let\u0027s just as a side exercise,"},{"Start":"02:14.389 ","End":"02:19.520","Text":"see what would happen if we multiply this numerator here by its conjugate."},{"Start":"02:19.520 ","End":"02:21.260","Text":"In this case, the A minus B,"},{"Start":"02:21.260 ","End":"02:24.615","Text":"this is A minus B will multiply it by A plus B."},{"Start":"02:24.615 ","End":"02:25.785","Text":"Let\u0027s see what happens."},{"Start":"02:25.785 ","End":"02:34.805","Text":"Square root of 2x plus 1 minus the square root of x plus 5."},{"Start":"02:34.805 ","End":"02:36.020","Text":"I\u0027m going to multiply,"},{"Start":"02:36.020 ","End":"02:37.610","Text":"so I\u0027ll need brackets,"},{"Start":"02:37.610 ","End":"02:40.465","Text":"and for the other, for the conjugate."},{"Start":"02:40.465 ","End":"02:42.775","Text":"Square root, same thing,"},{"Start":"02:42.775 ","End":"02:45.000","Text":"2x plus 1,"},{"Start":"02:45.000 ","End":"02:47.310","Text":"but this time with a plus in the middle,"},{"Start":"02:47.310 ","End":"02:52.130","Text":"and again, square root of x plus 5."},{"Start":"02:52.130 ","End":"02:54.260","Text":"Let\u0027s see what this gives us."},{"Start":"02:54.260 ","End":"02:56.300","Text":"If we look at this, A minus B,"},{"Start":"02:56.300 ","End":"02:58.690","Text":"A plus B is A squared minus B squared,"},{"Start":"02:58.690 ","End":"03:01.925","Text":"so our A squared is just the square root squared,"},{"Start":"03:01.925 ","End":"03:06.190","Text":"so this is just 2x plus 1,"},{"Start":"03:06.190 ","End":"03:08.040","Text":"which was the first,"},{"Start":"03:08.040 ","End":"03:10.415","Text":"minus same thing here."},{"Start":"03:10.415 ","End":"03:16.235","Text":"Just take up the square root minus x plus 5 with no square root,"},{"Start":"03:16.235 ","End":"03:25.110","Text":"and we just collect terms 2x minus x is x and 1 minus 5 is minus 4."},{"Start":"03:25.110 ","End":"03:31.439","Text":"This is just equal to x minus 4."},{"Start":"03:31.439 ","End":"03:35.765","Text":"Now we want to see how this is going to help us in finding our limit,"},{"Start":"03:35.765 ","End":"03:39.245","Text":"the limit, let me just write the exercise again,"},{"Start":"03:39.245 ","End":"03:41.450","Text":"x goes to 4."},{"Start":"03:41.450 ","End":"03:45.450","Text":"What we\u0027re going to do is to multiply,"},{"Start":"03:45.450 ","End":"03:48.680","Text":"we want to multiply the numerator by its conjugate,"},{"Start":"03:48.680 ","End":"03:50.300","Text":"which will give us a nice expressions."},{"Start":"03:50.300 ","End":"03:54.830","Text":"We multiply by square root of"},{"Start":"03:54.830 ","End":"04:02.385","Text":"2x plus 1 plus square root of x plus 5."},{"Start":"04:02.385 ","End":"04:05.540","Text":"I can\u0027t go ahead multiplying by whatever I want,"},{"Start":"04:05.540 ","End":"04:07.039","Text":"because I\u0027ve changed the exercise."},{"Start":"04:07.039 ","End":"04:13.140","Text":"But if I multiply top and bottom by the same thing,"},{"Start":"04:13.180 ","End":"04:18.170","Text":"this second fraction as a whole is just equal to 1,"},{"Start":"04:18.170 ","End":"04:21.320","Text":"A over A is 1, whatever A is."},{"Start":"04:21.320 ","End":"04:23.945","Text":"We haven\u0027t changed anything,"},{"Start":"04:23.945 ","End":"04:26.720","Text":"and we can remember to multiply fractions,"},{"Start":"04:26.720 ","End":"04:30.140","Text":"we multiply numerators, and we multiply denominators."},{"Start":"04:30.140 ","End":"04:37.140","Text":"What we get is the limit as x approaches 4 of,"},{"Start":"04:37.140 ","End":"04:40.640","Text":"this times this is an exercise we\u0027ve already done up here."},{"Start":"04:40.640 ","End":"04:45.625","Text":"The whole numerator just is x minus 4,"},{"Start":"04:45.625 ","End":"04:49.790","Text":"and on the denominator we have this x minus 4,"},{"Start":"04:49.790 ","End":"04:51.320","Text":"let me write that in brackets"},{"Start":"04:51.320 ","End":"04:53.815","Text":"because the denominator is like a bracket,"},{"Start":"04:53.815 ","End":"04:57.095","Text":"times whatever is left here,"},{"Start":"04:57.095 ","End":"05:07.550","Text":"which is the square root of 2x plus 1 plus the square root of x plus 5."},{"Start":"05:07.550 ","End":"05:11.720","Text":"Well, we are lucky that something cancels,"},{"Start":"05:11.720 ","End":"05:14.780","Text":"the x minus 4 cancels with the x minus 4,"},{"Start":"05:14.780 ","End":"05:17.845","Text":"leaving just a 1 in the numerator."},{"Start":"05:17.845 ","End":"05:21.905","Text":"Now we have to find the limit of this function,"},{"Start":"05:21.905 ","End":"05:23.810","Text":"which is also elementary,"},{"Start":"05:23.810 ","End":"05:27.790","Text":"but in this case, we can substitute x equals 4,"},{"Start":"05:27.790 ","End":"05:30.235","Text":"and this equals,"},{"Start":"05:30.235 ","End":"05:32.765","Text":"wherever I see x I put 4."},{"Start":"05:32.765 ","End":"05:41.700","Text":"I\u0027m going to get 1 over the square root of 2 times 4 plus 1,"},{"Start":"05:41.890 ","End":"05:49.535","Text":"plus the square root of 4 plus 5."},{"Start":"05:49.535 ","End":"05:51.155","Text":"Let\u0027s see what this equals,"},{"Start":"05:51.155 ","End":"05:53.300","Text":"square root of 9 is 3,"},{"Start":"05:53.300 ","End":"05:55.460","Text":"and this denominator is 3,"},{"Start":"05:55.460 ","End":"05:59.445","Text":"and the square root of 4 plus 5 is also 9,"},{"Start":"05:59.445 ","End":"06:01.545","Text":"square root of 9 is 3."},{"Start":"06:01.545 ","End":"06:04.050","Text":"We have 1 over 3 plus 3,"},{"Start":"06:04.050 ","End":"06:09.430","Text":"1/6, and that\u0027s our answer."}],"ID":1548},{"Watched":false,"Name":"Exercise 6","Duration":"8m 9s","ChapterTopicVideoID":1537,"CourseChapterTopicPlaylistID":65361,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1537.jpeg","UploadDate":"2014-10-22T04:07:20.3800000","DurationForVideoObject":"PT8M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.030","Text":"In this exercise, we have to find the limit"},{"Start":"00:03.030 ","End":"00:09.240","Text":"as x tends to 1 of the function 2 minus square root of 3x plus 1"},{"Start":"00:09.240 ","End":"00:12.990","Text":"over 1 minus the square root of 2x minus 1."},{"Start":"00:12.990 ","End":"00:14.925","Text":"Now, this is an elementary function,"},{"Start":"00:14.925 ","End":"00:16.785","Text":"and in such cases,"},{"Start":"00:16.785 ","End":"00:20.145","Text":"usually we can just substitute x equals 1."},{"Start":"00:20.145 ","End":"00:22.185","Text":"However, in our case,"},{"Start":"00:22.185 ","End":"00:23.565","Text":"if we try doing that,"},{"Start":"00:23.565 ","End":"00:29.445","Text":"we would get twice 1 minus 1 is 1 square root of 1 is 1,"},{"Start":"00:29.445 ","End":"00:31.620","Text":"1 minus 1 is 0."},{"Start":"00:31.620 ","End":"00:36.410","Text":"In the numerator, 3 times 1 plus 1 is 4."},{"Start":"00:36.410 ","End":"00:37.880","Text":"Square root of 4 is 2."},{"Start":"00:37.880 ","End":"00:39.775","Text":"2 minus 2 is 0."},{"Start":"00:39.775 ","End":"00:43.430","Text":"We are in a 0 over 0 situation."},{"Start":"00:43.430 ","End":"00:47.075","Text":"Actually we can\u0027t substitute x equals 1 unfortunately."},{"Start":"00:47.075 ","End":"00:49.060","Text":"We\u0027re going to have to use other techniques."},{"Start":"00:49.060 ","End":"00:50.315","Text":"Now, taking a look at it,"},{"Start":"00:50.315 ","End":"00:51.810","Text":"we see that there are square roots,"},{"Start":"00:51.810 ","End":"00:56.015","Text":"there are terms with pluses and minuses and some have square roots."},{"Start":"00:56.015 ","End":"01:00.695","Text":"That usually indicates that we should use the technique of conjugates."},{"Start":"01:00.695 ","End":"01:03.955","Text":"I\u0027d like to remind you what a conjugate is."},{"Start":"01:03.955 ","End":"01:08.965","Text":"In general, if I have an expression made up of 2 terms,"},{"Start":"01:08.965 ","End":"01:10.690","Text":"like A plus B,"},{"Start":"01:10.690 ","End":"01:15.740","Text":"its conjugate is A minus B and vice versa."},{"Start":"01:15.740 ","End":"01:19.130","Text":"If I have something of the form A minus B,"},{"Start":"01:19.130 ","End":"01:21.350","Text":"its conjugate is A plus B."},{"Start":"01:21.350 ","End":"01:24.050","Text":"These 2 are conjugate to each other."},{"Start":"01:24.050 ","End":"01:30.320","Text":"The useful thing about them in connection with square roots is that if we multiply them"},{"Start":"01:30.320 ","End":"01:38.825","Text":"A plus B times A minus B is 1 of those formerly called a difference of squares,"},{"Start":"01:38.825 ","End":"01:42.815","Text":"where this is equal to A squared minus B squared."},{"Start":"01:42.815 ","End":"01:48.470","Text":"If A or B or both possibly have square roots after the square,"},{"Start":"01:48.470 ","End":"01:50.420","Text":"they no longer have the square root."},{"Start":"01:50.420 ","End":"01:53.345","Text":"Let\u0027s see how this helps us in our case."},{"Start":"01:53.345 ","End":"01:55.805","Text":"Both in numerator and in denominator,"},{"Start":"01:55.805 ","End":"01:57.365","Text":"we have square roots."},{"Start":"01:57.365 ","End":"02:00.770","Text":"It might be useful to do conjugates twice,"},{"Start":"02:00.770 ","End":"02:03.365","Text":"once for the numerator and once for the denominator."},{"Start":"02:03.365 ","End":"02:06.230","Text":"Let\u0027s do this as a pair of side exercises."},{"Start":"02:06.230 ","End":"02:09.890","Text":"We\u0027ll multiply this by its conjugate and then this by its conjugate,"},{"Start":"02:09.890 ","End":"02:11.825","Text":"and later we\u0027ll tie it all in."},{"Start":"02:11.825 ","End":"02:17.720","Text":"Let\u0027s take first of all the 2 minus the square root of"},{"Start":"02:17.720 ","End":"02:23.450","Text":"3x plus 1 from the numerator and multiply it by its conjugate,"},{"Start":"02:23.450 ","End":"02:24.695","Text":"it\u0027s a minus here,"},{"Start":"02:24.695 ","End":"02:31.325","Text":"so we now need a plus square root of 3x plus 1."},{"Start":"02:31.325 ","End":"02:34.265","Text":"This equals according to this formula,"},{"Start":"02:34.265 ","End":"02:37.505","Text":"first 1 squared minus the second 1 squared."},{"Start":"02:37.505 ","End":"02:39.204","Text":"It\u0027s the 2 squared,"},{"Start":"02:39.204 ","End":"02:42.740","Text":"which is 4 minus the second 1,"},{"Start":"02:42.740 ","End":"02:49.510","Text":"which is now without the square roots, 3x plus 1."},{"Start":"02:49.520 ","End":"02:52.445","Text":"If we simplify this,"},{"Start":"02:52.445 ","End":"02:56.420","Text":"we get 3 minus 3x."},{"Start":"02:56.420 ","End":"03:00.950","Text":"That\u0027s what happens when we multiply the numerator by its conjugate."},{"Start":"03:00.950 ","End":"03:04.265","Text":"Now let\u0027s do the same thing on the denominator which is here."},{"Start":"03:04.265 ","End":"03:10.720","Text":"We\u0027ll take the denominator 1 minus the square root of 2x minus 1,"},{"Start":"03:10.720 ","End":"03:13.005","Text":"and multiply it by its conjugate."},{"Start":"03:13.005 ","End":"03:17.870","Text":"We turn the minus into a plus and we get a similar way."},{"Start":"03:17.870 ","End":"03:20.585","Text":"The first 1 squared minus second 1 squared."},{"Start":"03:20.585 ","End":"03:25.340","Text":"1 squared is 1 less 2x minus"},{"Start":"03:25.340 ","End":"03:31.900","Text":"1 and what we get is 2 minus 2x."},{"Start":"03:33.050 ","End":"03:37.215","Text":"These are side exercises meanwhile."},{"Start":"03:37.215 ","End":"03:39.950","Text":"Let\u0027s see how we can tie them in here."},{"Start":"03:39.950 ","End":"03:44.795","Text":"I\u0027ll continue just by copying the exercise as it was over here."},{"Start":"03:44.795 ","End":"03:47.060","Text":"Now what I\u0027d like to do,"},{"Start":"03:47.060 ","End":"03:50.300","Text":"I\u0027m going to spoil it and then fix it again."},{"Start":"03:50.300 ","End":"03:52.820","Text":"What I\u0027d like to do is multiply this thing,"},{"Start":"03:52.820 ","End":"03:55.385","Text":"the numerator by its conjugate."},{"Start":"03:55.385 ","End":"04:00.540","Text":"Because I know that will give me something nicer without square roots."},{"Start":"04:00.540 ","End":"04:06.180","Text":"2 plus a square root of 3x plus 1."},{"Start":"04:06.180 ","End":"04:12.110","Text":"What I\u0027d like to do also is multiply the denominator by its conjugate,"},{"Start":"04:12.110 ","End":"04:13.790","Text":"like in the second line,"},{"Start":"04:13.790 ","End":"04:20.630","Text":"and that\u0027s 1 plus the square root of 2x minus 1."},{"Start":"04:20.630 ","End":"04:22.340","Text":"That\u0027s what I\u0027d like to do."},{"Start":"04:22.340 ","End":"04:25.430","Text":"But if I multiply this thing by something and then I\u0027ve ruined it,"},{"Start":"04:25.430 ","End":"04:27.560","Text":"so I have to now fix it."},{"Start":"04:27.560 ","End":"04:30.950","Text":"Let me just show you as a side exercise."},{"Start":"04:30.950 ","End":"04:40.025","Text":"If I take a fraction a over b and then multiply it by b over a,"},{"Start":"04:40.025 ","End":"04:42.560","Text":"the b cancels with b, a with a,"},{"Start":"04:42.560 ","End":"04:45.230","Text":"this is equal to 1."},{"Start":"04:45.230 ","End":"04:48.125","Text":"Essentially this is just the inverse fraction."},{"Start":"04:48.125 ","End":"04:52.400","Text":"In our case, in order to save the exercise and not to spoil it,"},{"Start":"04:52.400 ","End":"04:56.790","Text":"with this, If I now multiply by the opposite, in other words,"},{"Start":"04:56.790 ","End":"05:00.240","Text":"I\u0027ll put this on the numerator and here,"},{"Start":"05:00.240 ","End":"05:06.255","Text":"2 plus the square root of 3x plus 1,"},{"Start":"05:06.255 ","End":"05:09.980","Text":"then I\u0027ll be in a situation where I\u0027ve taken the original fraction,"},{"Start":"05:09.980 ","End":"05:14.330","Text":"multiplied it by sum a over b and then by sum b over a."},{"Start":"05:14.330 ","End":"05:17.360","Text":"This whole thing together is equal to 1,"},{"Start":"05:17.360 ","End":"05:19.975","Text":"so I haven\u0027t changed anything."},{"Start":"05:19.975 ","End":"05:27.090","Text":"This equals the limit as x goes to 1."},{"Start":"05:27.090 ","End":"05:31.729","Text":"Now, this times this we\u0027ve already done in this exercise,"},{"Start":"05:31.729 ","End":"05:34.510","Text":"and that\u0027s 3 minus 3x."},{"Start":"05:34.510 ","End":"05:39.500","Text":"This with this gives us what we did here,"},{"Start":"05:39.500 ","End":"05:43.260","Text":"which is 2 minus 2x."},{"Start":"05:44.510 ","End":"05:47.850","Text":"Now this times this is this,"},{"Start":"05:47.850 ","End":"05:49.260","Text":"I\u0027ll just put it in brackets,"},{"Start":"05:49.260 ","End":"05:51.855","Text":"the same thing with this pair in brackets."},{"Start":"05:51.855 ","End":"05:56.915","Text":"What we\u0027re left with is the numerator and denominator from the last of the 3,"},{"Start":"05:56.915 ","End":"06:04.335","Text":"which is brackets 1 plus the square root of 2x minus 1."},{"Start":"06:04.335 ","End":"06:12.720","Text":"Here, 2 plus the square root of 3x plus 1."},{"Start":"06:12.720 ","End":"06:16.530","Text":"Usually at this point, something cancels."},{"Start":"06:16.530 ","End":"06:19.025","Text":"We\u0027re looking for something to cancel."},{"Start":"06:19.025 ","End":"06:25.430","Text":"If you notice, we can take 3 out of this and 2 out of this,"},{"Start":"06:25.430 ","End":"06:31.100","Text":"and what we\u0027ll get is the limit x tends to 1,"},{"Start":"06:31.100 ","End":"06:36.860","Text":"3 times 1 minus x times the other piece,"},{"Start":"06:36.860 ","End":"06:38.165","Text":"I\u0027ll fill it in in a minute,"},{"Start":"06:38.165 ","End":"06:43.860","Text":"and here twice, 1 minus x times the last bit."},{"Start":"06:43.860 ","End":"06:48.690","Text":"1 plus the square root of 2x minus 1,"},{"Start":"06:48.690 ","End":"06:54.720","Text":"2 plus the square root of 3x plus 1."},{"Start":"06:54.720 ","End":"06:58.145","Text":"Now x tends to 1. x is not equal to 1,"},{"Start":"06:58.145 ","End":"07:00.440","Text":"so 1 minus x is not 0,"},{"Start":"07:00.440 ","End":"07:02.135","Text":"and we can cancel."},{"Start":"07:02.135 ","End":"07:07.220","Text":"After this cancellation, this function that\u0027s left is still elementary,"},{"Start":"07:07.220 ","End":"07:12.005","Text":"and this time we can substitute x equals 1 to get the answer."},{"Start":"07:12.005 ","End":"07:15.875","Text":"I\u0027m taking this expression and putting in x equals 1."},{"Start":"07:15.875 ","End":"07:18.070","Text":"What I get is"},{"Start":"07:18.070 ","End":"07:28.665","Text":"3 times 1 plus square root of 2 times 1 minus 1 over 2 from here,"},{"Start":"07:28.665 ","End":"07:31.500","Text":"2 plus, now x again is 1,"},{"Start":"07:31.500 ","End":"07:37.960","Text":"the square root of 3 times 1 plus 1."},{"Start":"07:38.710 ","End":"07:41.075","Text":"Let\u0027s see what we\u0027ve got here."},{"Start":"07:41.075 ","End":"07:45.360","Text":"This is 3, this is 2."},{"Start":"07:45.650 ","End":"07:49.410","Text":"Here we have 1 plus 1."},{"Start":"07:49.410 ","End":"07:55.065","Text":"Square root is 1, 2 plus 2."},{"Start":"07:55.065 ","End":"07:59.105","Text":"This is 6 over 8,"},{"Start":"07:59.105 ","End":"08:07.705","Text":"which we can simplify top and bottom divide by 2 to 3/4."},{"Start":"08:07.705 ","End":"08:10.540","Text":"That\u0027s the end of the exercise."}],"ID":1549},{"Watched":false,"Name":"Exercise 7","Duration":"4m 6s","ChapterTopicVideoID":1538,"CourseChapterTopicPlaylistID":65361,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1538.jpeg","UploadDate":"2014-10-22T04:07:39.3100000","DurationForVideoObject":"PT4M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.915","Text":"In this exercise, we want to find the limit as x goes to 1 of the function,"},{"Start":"00:06.915 ","End":"00:10.680","Text":"1 minus the cube root of x over 1 minus x."},{"Start":"00:10.680 ","End":"00:17.730","Text":"Now, sometimes we can find the limit just by substituting x equals 1 into this function."},{"Start":"00:17.730 ","End":"00:20.775","Text":"But unfortunately it doesn\u0027t work here."},{"Start":"00:20.775 ","End":"00:22.620","Text":"Because if we put here x equals 1,"},{"Start":"00:22.620 ","End":"00:25.650","Text":"we get 0 and if we put x equals 1 here,"},{"Start":"00:25.650 ","End":"00:28.335","Text":"we also get 1 minus 1 is 0,"},{"Start":"00:28.335 ","End":"00:32.925","Text":"so this is one of those exercises of the form 0/0."},{"Start":"00:32.925 ","End":"00:35.400","Text":"So we\u0027re going to have to use some other techniques."},{"Start":"00:35.400 ","End":"00:38.835","Text":"Now, if this was a square root and not a cube root,"},{"Start":"00:38.835 ","End":"00:41.185","Text":"we\u0027d go with the method of the conjugates."},{"Start":"00:41.185 ","End":"00:45.320","Text":"There is something similar for cube roots as opposed to square roots."},{"Start":"00:45.320 ","End":"00:47.360","Text":"It\u0027s similar to conjugate,"},{"Start":"00:47.360 ","End":"00:51.670","Text":"and I\u0027d like to write some formulas from algebra for you."},{"Start":"00:51.670 ","End":"00:53.880","Text":"What I\u0027ve written is, first of all,"},{"Start":"00:53.880 ","End":"00:57.260","Text":"a reminder of what we used to do in the case of conjugates,"},{"Start":"00:57.260 ","End":"00:59.490","Text":"A minus B and A plus B are conjugates,"},{"Start":"00:59.490 ","End":"01:00.665","Text":"and if I multiply,"},{"Start":"01:00.665 ","End":"01:02.930","Text":"I get a difference of squares."},{"Start":"01:02.930 ","End":"01:06.890","Text":"There is a similar formula in algebra called a difference of cubes."},{"Start":"01:06.890 ","End":"01:09.500","Text":"Essentially, what I\u0027ve written here is that"},{"Start":"01:09.500 ","End":"01:12.230","Text":"if I have A minus B and multiply it by this thing,"},{"Start":"01:12.230 ","End":"01:14.030","Text":"A squared plus AB plus B squared,"},{"Start":"01:14.030 ","End":"01:15.515","Text":"I get a difference of cubes."},{"Start":"01:15.515 ","End":"01:18.545","Text":"This will help me to get rid of cube roots."},{"Start":"01:18.545 ","End":"01:21.455","Text":"Let\u0027s see how it works in our case."},{"Start":"01:21.455 ","End":"01:26.025","Text":"Look at the numerator first and think of this as my A minus B."},{"Start":"01:26.025 ","End":"01:30.105","Text":"So 1 minus the cube root of x."},{"Start":"01:30.105 ","End":"01:33.320","Text":"Instead of multiplying as we did in the square root case,"},{"Start":"01:33.320 ","End":"01:37.250","Text":"with the conjugate, I\u0027m going to multiply by the equivalent of what this is here,"},{"Start":"01:37.250 ","End":"01:40.465","Text":"so we get A squared,"},{"Start":"01:40.465 ","End":"01:43.335","Text":"which is 1 plus AB,"},{"Start":"01:43.335 ","End":"01:46.095","Text":"which is 1 times the cube root of x,"},{"Start":"01:46.095 ","End":"01:49.970","Text":"and B squared, which is the cube root of x squared."},{"Start":"01:49.970 ","End":"01:52.490","Text":"If I multiply this, what I get,"},{"Start":"01:52.490 ","End":"01:53.720","Text":"according to this formula,"},{"Start":"01:53.720 ","End":"01:55.820","Text":"is A cubed minus B cubed,"},{"Start":"01:55.820 ","End":"01:58.565","Text":"which is 1 cubed is 1,"},{"Start":"01:58.565 ","End":"02:02.660","Text":"and B cubed is the cube root cubed,"},{"Start":"02:02.660 ","End":"02:04.725","Text":"which is just the thing itself,"},{"Start":"02:04.725 ","End":"02:07.395","Text":"so I get 1 minus x."},{"Start":"02:07.395 ","End":"02:11.235","Text":"Let\u0027s see how this helps me in this exercise."},{"Start":"02:11.235 ","End":"02:17.450","Text":"I start off again by writing the limit and just copying of 1"},{"Start":"02:17.450 ","End":"02:25.714","Text":"minus the cube root of x over 1 minus x."},{"Start":"02:25.714 ","End":"02:31.190","Text":"What I\u0027d like to do is since multiplying by this gives such a simple expression,"},{"Start":"02:31.190 ","End":"02:36.560","Text":"I\u0027d like to multiply this by 1 plus"},{"Start":"02:36.560 ","End":"02:44.675","Text":"the cube root of x plus the cube root of x squared."},{"Start":"02:44.675 ","End":"02:48.320","Text":"I can\u0027t just go ahead and multiply because I\u0027m changing it."},{"Start":"02:48.320 ","End":"02:53.120","Text":"However, if I write the same thing on the denominator,"},{"Start":"02:53.120 ","End":"02:56.210","Text":"this fraction is of the form A/A,"},{"Start":"02:56.210 ","End":"02:58.415","Text":"same numerator as denominator,"},{"Start":"02:58.415 ","End":"03:00.770","Text":"and such a thing is equal to 1."},{"Start":"03:00.770 ","End":"03:03.950","Text":"I can certainly multiply by 1 without changing anything."},{"Start":"03:03.950 ","End":"03:06.740","Text":"Though continuing, do a bit of the algebra,"},{"Start":"03:06.740 ","End":"03:11.180","Text":"we get the limit as x goes to 1."},{"Start":"03:11.180 ","End":"03:13.010","Text":"This times this,"},{"Start":"03:13.010 ","End":"03:15.230","Text":"we\u0027ve already done over here, in other words,"},{"Start":"03:15.230 ","End":"03:19.070","Text":"we\u0027ve computed the numerator to be 1 minus x."},{"Start":"03:19.070 ","End":"03:21.545","Text":"Now in the denominator,"},{"Start":"03:21.545 ","End":"03:27.050","Text":"I just leave this 1 minus x as it is and copy this bit here."},{"Start":"03:27.050 ","End":"03:29.330","Text":"What I notice is 1 minus x here,"},{"Start":"03:29.330 ","End":"03:31.795","Text":"the 1 minus x here, and they cancel."},{"Start":"03:31.795 ","End":"03:34.905","Text":"Of course that leaves a 1 in the numerator."},{"Start":"03:34.905 ","End":"03:39.140","Text":"Now I\u0027ve got to find the limit as x goes to 1 of a much simpler function by"},{"Start":"03:39.140 ","End":"03:43.780","Text":"which I mean that we can substitute x equals 1 in this case where we couldn\u0027t before."},{"Start":"03:43.780 ","End":"03:48.155","Text":"All we have to do to evaluate this limit is to let x equal 1."},{"Start":"03:48.155 ","End":"03:58.165","Text":"We get just almost copying 1 over the 1 from here plus the cube root of 1 is 1,"},{"Start":"03:58.165 ","End":"04:01.590","Text":"and the cube root of 1 squared is also 1,"},{"Start":"04:01.590 ","End":"04:07.180","Text":"so the final answer is 1/3. That\u0027s it."}],"ID":1550}],"Thumbnail":null,"ID":65361},{"Name":"Technique 4 - Function Tends to Infinity","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Function Tends to Infinity","Duration":"9m 43s","ChapterTopicVideoID":8251,"CourseChapterTopicPlaylistID":65362,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8251.jpeg","UploadDate":"2019-10-29T04:21:58.2230000","DurationForVideoObject":"PT9M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.505","Text":"In this clip, we introduce technique number 4 for finding limits of functions."},{"Start":"00:05.505 ","End":"00:08.775","Text":"This technique is functions that tend to infinity."},{"Start":"00:08.775 ","End":"00:11.700","Text":"At last, we talk about the number infinity,"},{"Start":"00:11.700 ","End":"00:13.740","Text":"which in fact is not even a number at all."},{"Start":"00:13.740 ","End":"00:19.350","Text":"It\u0027s a symbol for some quantity or number that grows without bound."},{"Start":"00:19.350 ","End":"00:22.110","Text":"In fact, there\u0027s also a minus infinity,"},{"Start":"00:22.110 ","End":"00:24.380","Text":"which is something which shrinks without bounds."},{"Start":"00:24.380 ","End":"00:27.140","Text":"When I say shrink, I mean get more and more negative,"},{"Start":"00:27.140 ","End":"00:28.350","Text":"not more and more tiny."},{"Start":"00:28.350 ","End":"00:30.090","Text":"When do we use such a technique?"},{"Start":"00:30.090 ","End":"00:35.805","Text":"When we have a limit of the form not 0 over tends to 0,"},{"Start":"00:35.805 ","End":"00:39.119","Text":"then we get a function that tends to infinity."},{"Start":"00:39.119 ","End":"00:40.690","Text":"I want to be more precise."},{"Start":"00:40.690 ","End":"00:44.690","Text":"This limit will be when x goes to some specific value,"},{"Start":"00:44.690 ","End":"00:52.205","Text":"say a and what I want is for this not 0 to be not 0 at x equals a,"},{"Start":"00:52.205 ","End":"00:56.555","Text":"and the tends to 0 as x tends to a,"},{"Start":"00:56.555 ","End":"00:58.220","Text":"just to be a bit more precise."},{"Start":"00:58.220 ","End":"01:00.590","Text":"This is a bit different from what we\u0027re familiar with."},{"Start":"01:00.590 ","End":"01:04.005","Text":"We\u0027re familiar with tends to 0 over tends to 0."},{"Start":"01:04.005 ","End":"01:07.295","Text":"Now we have not 0 over tends to 0."},{"Start":"01:07.295 ","End":"01:10.670","Text":"Let\u0027s do some examples to see what this really means."},{"Start":"01:10.670 ","End":"01:12.004","Text":"For the first example,"},{"Start":"01:12.004 ","End":"01:14.180","Text":"we\u0027ll take the simplest case;"},{"Start":"01:14.180 ","End":"01:20.015","Text":"the limit as x tends to 0 of 1 over x."},{"Start":"01:20.015 ","End":"01:23.195","Text":"Now here clearly the numerator is 1,"},{"Start":"01:23.195 ","End":"01:25.745","Text":"so it\u0027s definitely not 0 anywhere."},{"Start":"01:25.745 ","End":"01:30.560","Text":"The denominator x tends to 0 as x tends to 0."},{"Start":"01:30.560 ","End":"01:35.465","Text":"That\u0027s obvious. Here we have non 0 over tends to 0."},{"Start":"01:35.465 ","End":"01:44.670","Text":"Another example is the limit as x tends to 1 of x plus 2 over x minus 1."},{"Start":"01:44.670 ","End":"01:46.310","Text":"If we examined this again,"},{"Start":"01:46.310 ","End":"01:49.310","Text":"we\u0027ll see that technique number 4 is useful because"},{"Start":"01:49.310 ","End":"01:55.145","Text":"the numerator x plus 2 is definitely not 0 when x equals 1, it\u0027s 3."},{"Start":"01:55.145 ","End":"01:56.630","Text":"I\u0027m not saying it can\u0027t be 0."},{"Start":"01:56.630 ","End":"01:59.090","Text":"Sometimes it could be 0 when x is minus 2,"},{"Start":"01:59.090 ","End":"02:01.010","Text":"but we\u0027re not talking about minus 2,"},{"Start":"02:01.010 ","End":"02:02.090","Text":"we\u0027re talking about 1."},{"Start":"02:02.090 ","End":"02:03.470","Text":"Here it\u0027s definitely not 0."},{"Start":"02:03.470 ","End":"02:08.720","Text":"On the other hand, the denominator x minus 1 does go to 0 when x tends to 1."},{"Start":"02:08.720 ","End":"02:10.970","Text":"Quite clearly, x gets closer and closer to 1,"},{"Start":"02:10.970 ","End":"02:13.730","Text":"x minus 1 gets closer and closer to 0."},{"Start":"02:13.730 ","End":"02:15.650","Text":"This is a second example."},{"Start":"02:15.650 ","End":"02:17.915","Text":"Now let\u0027s look at a third example."},{"Start":"02:17.915 ","End":"02:23.050","Text":"Limit as x tends to 4 of x"},{"Start":"02:23.050 ","End":"02:28.670","Text":"squared plus x plus 1 over x minus 4."},{"Start":"02:28.670 ","End":"02:32.570","Text":"Once again, we look at the numerator to make sure it\u0027s not 0."},{"Start":"02:32.570 ","End":"02:33.920","Text":"If I put 4 in here,"},{"Start":"02:33.920 ","End":"02:36.770","Text":"4 squared plus 4 plus 1 is actually 21,"},{"Start":"02:36.770 ","End":"02:39.485","Text":"which is definitely not 0."},{"Start":"02:39.485 ","End":"02:42.680","Text":"Whereas the denominator definitely does go to"},{"Start":"02:42.680 ","End":"02:46.450","Text":"0 and something very important I have to add;"},{"Start":"02:46.450 ","End":"02:48.990","Text":"when we use this technique i.e.,"},{"Start":"02:48.990 ","End":"02:51.480","Text":"the not 0 over tends to 0,"},{"Start":"02:51.480 ","End":"02:56.990","Text":"it\u0027s very important to distinguish between the left limit and the right limit."},{"Start":"02:56.990 ","End":"02:58.580","Text":"This is because it makes"},{"Start":"02:58.580 ","End":"03:02.510","Text":"a very big difference if we\u0027re dividing by something that\u0027s very,"},{"Start":"03:02.510 ","End":"03:05.015","Text":"very close to 0 but positive,"},{"Start":"03:05.015 ","End":"03:08.555","Text":"or very, very close to 0 but negative."},{"Start":"03:08.555 ","End":"03:11.240","Text":"We\u0027ll see this in more detail later on."},{"Start":"03:11.240 ","End":"03:13.730","Text":"But first, let me put it down in writing."},{"Start":"03:13.730 ","End":"03:15.255","Text":"In such cases,"},{"Start":"03:15.255 ","End":"03:18.915","Text":"and I mean the non 0 over tends to 0,"},{"Start":"03:18.915 ","End":"03:22.520","Text":"we have to separate the limit into the limit from the left"},{"Start":"03:22.520 ","End":"03:26.150","Text":"or left limit and the limit from the right or the right limit."},{"Start":"03:26.150 ","End":"03:28.610","Text":"Only if these 2 are equal,"},{"Start":"03:28.610 ","End":"03:30.425","Text":"does the function have a limit."},{"Start":"03:30.425 ","End":"03:32.750","Text":"Again, we\u0027ll see this in the examples."},{"Start":"03:32.750 ","End":"03:35.315","Text":"Let\u0027s illustrate with the first example above,"},{"Start":"03:35.315 ","End":"03:42.455","Text":"which was the limit as x goes to 0 of 1 over x. I want to know what that is."},{"Start":"03:42.455 ","End":"03:47.270","Text":"I need the limit as x goes to 0 from the right,"},{"Start":"03:47.270 ","End":"03:52.010","Text":"you write it with a plus here and limit as x goes to 0"},{"Start":"03:52.010 ","End":"03:56.760","Text":"from the left of the same thing and I\u0027ll see what each of these is."},{"Start":"03:56.760 ","End":"03:58.610","Text":"I\u0027ve a bit of coloring here."},{"Start":"03:58.610 ","End":"03:59.810","Text":"Let\u0027s do the first one;"},{"Start":"03:59.810 ","End":"04:06.290","Text":"the x goes to 0 from the right and make a little table here of values x,"},{"Start":"04:06.290 ","End":"04:10.190","Text":"y, where y equals the 1 over x."},{"Start":"04:10.190 ","End":"04:14.060","Text":"If I take something like x equals something small,"},{"Start":"04:14.060 ","End":"04:15.665","Text":"1 over 100,"},{"Start":"04:15.665 ","End":"04:18.470","Text":"then y, which is 1 over x, will be 100."},{"Start":"04:18.470 ","End":"04:20.380","Text":"If I take x even smaller,"},{"Start":"04:20.380 ","End":"04:21.930","Text":"1 over, I don\u0027t know,"},{"Start":"04:21.930 ","End":"04:26.930","Text":"50,000, then y will be 50,000 getting larger."},{"Start":"04:26.930 ","End":"04:28.940","Text":"If I take something very, very small,"},{"Start":"04:28.940 ","End":"04:30.215","Text":"1 over 1,000,000,"},{"Start":"04:30.215 ","End":"04:32.060","Text":"then y is going to be 1,000,000."},{"Start":"04:32.060 ","End":"04:35.165","Text":"We can see that as x gets smaller and smaller,"},{"Start":"04:35.165 ","End":"04:37.729","Text":"but from the right in other words positive,"},{"Start":"04:37.729 ","End":"04:42.695","Text":"then y gets larger and larger without bound, it gets large."},{"Start":"04:42.695 ","End":"04:45.800","Text":"That\u0027s what we mean when we say goes to infinity."},{"Start":"04:45.800 ","End":"04:49.030","Text":"It gets large and is not bounded in size."},{"Start":"04:49.030 ","End":"04:52.775","Text":"Gets larger than anything you want and keep getting larger and larger."},{"Start":"04:52.775 ","End":"05:00.740","Text":"I\u0027m going to write in this case that the limit as x goes to 0 of 1 over x is infinity."},{"Start":"05:00.740 ","End":"05:05.405","Text":"Now how about when x goes to 0 from the left?"},{"Start":"05:05.405 ","End":"05:08.030","Text":"Which means that it\u0027s slightly negative."},{"Start":"05:08.030 ","End":"05:09.455","Text":"It gets closer and closer to 0,"},{"Start":"05:09.455 ","End":"05:11.300","Text":"but through negative numbers."},{"Start":"05:11.300 ","End":"05:13.355","Text":"I don\u0027t even have to make a new table."},{"Start":"05:13.355 ","End":"05:15.125","Text":"I could just put minus here,"},{"Start":"05:15.125 ","End":"05:17.150","Text":"minus here, and minus here."},{"Start":"05:17.150 ","End":"05:19.070","Text":"When x is minus 100th,"},{"Start":"05:19.070 ","End":"05:20.870","Text":"y is going to be minus 100."},{"Start":"05:20.870 ","End":"05:23.315","Text":"This looks like it\u0027s getting very, very small,"},{"Start":"05:23.315 ","End":"05:27.030","Text":"meaning negative, large but negative if you like."},{"Start":"05:27.030 ","End":"05:30.200","Text":"The answer here is going to be minus infinity."},{"Start":"05:30.200 ","End":"05:34.010","Text":"Now what\u0027s happened is that there is a limit on the left,"},{"Start":"05:34.010 ","End":"05:36.770","Text":"at least in the incentive infinity or minus infinity,"},{"Start":"05:36.770 ","End":"05:38.465","Text":"but they\u0027re not equal."},{"Start":"05:38.465 ","End":"05:45.470","Text":"What I say in such a case is that the limit of 1 over x as a whole does not exist."},{"Start":"05:45.470 ","End":"05:49.010","Text":"Of course, if both of these had come out infinity,"},{"Start":"05:49.010 ","End":"05:51.620","Text":"then I would have written this limit as infinity,"},{"Start":"05:51.620 ","End":"05:53.960","Text":"or if both of these would come out minus infinity l"},{"Start":"05:53.960 ","End":"05:56.725","Text":"also would have written this limit as minus infinity."},{"Start":"05:56.725 ","End":"05:58.320","Text":"But here they didn\u0027t come out the same,"},{"Start":"05:58.320 ","End":"05:59.760","Text":"so there just is no limit."},{"Start":"05:59.760 ","End":"06:02.120","Text":"I would like to show you what happens graphically,"},{"Start":"06:02.120 ","End":"06:04.175","Text":"just to help you get an idea."},{"Start":"06:04.175 ","End":"06:05.900","Text":"Y equals 1 over x,"},{"Start":"06:05.900 ","End":"06:08.345","Text":"which consists of 2 parts."},{"Start":"06:08.345 ","End":"06:11.060","Text":"This part here and this part here."},{"Start":"06:11.060 ","End":"06:17.090","Text":"The idea is I take values that keep getting closer and closer to 0,"},{"Start":"06:17.090 ","End":"06:19.010","Text":"but from the positive side,"},{"Start":"06:19.010 ","End":"06:21.305","Text":"then above them, on the graph,"},{"Start":"06:21.305 ","End":"06:24.560","Text":"I\u0027m going to get values that are just going to"},{"Start":"06:24.560 ","End":"06:28.750","Text":"get larger and larger and larger towards infinity."},{"Start":"06:28.750 ","End":"06:32.255","Text":"If we take values that go to 0 from the left,"},{"Start":"06:32.255 ","End":"06:36.035","Text":"we\u0027ll get values that go to minus infinity."},{"Start":"06:36.035 ","End":"06:39.215","Text":"These 2 don\u0027t meet and it\u0027s not the same limit."},{"Start":"06:39.215 ","End":"06:40.580","Text":"Just by the way,"},{"Start":"06:40.580 ","End":"06:42.800","Text":"if you have a line such as the y-axis,"},{"Start":"06:42.800 ","End":"06:45.740","Text":"in this case, which the function tends towards,"},{"Start":"06:45.740 ","End":"06:49.180","Text":"gets closer and closer but never actually reaches,"},{"Start":"06:49.180 ","End":"06:51.900","Text":"then this line, vertical line like this,"},{"Start":"06:51.900 ","End":"06:54.045","Text":"is called a vertical asymptote."},{"Start":"06:54.045 ","End":"06:55.925","Text":"Onto another example."},{"Start":"06:55.925 ","End":"06:58.520","Text":"This time we\u0027ll take as an example,"},{"Start":"06:58.520 ","End":"07:01.250","Text":"the limit as x goes to 0,"},{"Start":"07:01.250 ","End":"07:05.480","Text":"not 1 over x, but this time 1 over x squared."},{"Start":"07:05.480 ","End":"07:11.565","Text":"As before, this is something which is not 0 over something that tends to 0."},{"Start":"07:11.565 ","End":"07:13.805","Text":"We split it up into 2 cases."},{"Start":"07:13.805 ","End":"07:18.890","Text":"We take the limit as x goes to 0 from the right,"},{"Start":"07:18.890 ","End":"07:20.450","Text":"I\u0027ll write that in a minute,"},{"Start":"07:20.450 ","End":"07:25.430","Text":"of 1 over x squared and the limit as x goes to 0 from the left,"},{"Start":"07:25.430 ","End":"07:26.795","Text":"I\u0027ll put that in a minute,"},{"Start":"07:26.795 ","End":"07:28.970","Text":"of 1 over x squared."},{"Start":"07:28.970 ","End":"07:31.265","Text":"I want to put a plus in blue,"},{"Start":"07:31.265 ","End":"07:33.935","Text":"but a minus in red."},{"Start":"07:33.935 ","End":"07:36.370","Text":"Just like before, we made a table."},{"Start":"07:36.370 ","End":"07:41.420","Text":"If we start out with some simple examples like x is something small,"},{"Start":"07:41.420 ","End":"07:43.940","Text":"maybe 1 over 100,"},{"Start":"07:43.940 ","End":"07:46.930","Text":"then we\u0027ll try 1 over 1,000,"},{"Start":"07:46.930 ","End":"07:50.550","Text":"then we\u0027ll try 1 over 10,000."},{"Start":"07:50.550 ","End":"07:54.425","Text":"We get the corresponding y is 1 over x squared,"},{"Start":"07:54.425 ","End":"07:56.240","Text":"so it\u0027s 1 over 100 squared."},{"Start":"07:56.240 ","End":"07:58.775","Text":"This is actually 10,000."},{"Start":"07:58.775 ","End":"08:06.010","Text":"1 over 1,000 is 1,000,000 and 1 over 10,000 is 100,000,000."},{"Start":"08:06.010 ","End":"08:08.885","Text":"This also gets large without bound,"},{"Start":"08:08.885 ","End":"08:10.930","Text":"but much quicker than the one before."},{"Start":"08:10.930 ","End":"08:15.255","Text":"We can easily see that this one is infinity."},{"Start":"08:15.255 ","End":"08:19.085","Text":"If we take the other limit on the left,"},{"Start":"08:19.085 ","End":"08:23.120","Text":"then all I have to do is add a minus in front of these."},{"Start":"08:23.120 ","End":"08:27.800","Text":"But notice that these don\u0027t change because also minus 100,"},{"Start":"08:27.800 ","End":"08:29.090","Text":"if I take 1 over x squared,"},{"Start":"08:29.090 ","End":"08:32.585","Text":"it\u0027s also 10,000 plus because of the squared."},{"Start":"08:32.585 ","End":"08:36.980","Text":"Here we also get infinity and that means that this"},{"Start":"08:36.980 ","End":"08:41.600","Text":"time we do have a limit here and this is equal to infinity."},{"Start":"08:41.600 ","End":"08:45.380","Text":"As before, I\u0027d also like to show you what happens graphically."},{"Start":"08:45.380 ","End":"08:48.230","Text":"I\u0027ll just do a real rough and ready sketch here."},{"Start":"08:48.230 ","End":"08:52.985","Text":"There\u0027s not going to be anything below the x-axis because everything\u0027s positive."},{"Start":"08:52.985 ","End":"08:55.280","Text":"This one goes something like this,"},{"Start":"08:55.280 ","End":"08:58.880","Text":"I happen to know, and this one goes also just the mirror image of it."},{"Start":"08:58.880 ","End":"09:00.955","Text":"Notice that it\u0027s an even function."},{"Start":"09:00.955 ","End":"09:05.165","Text":"When we go to 0 from the positive side,"},{"Start":"09:05.165 ","End":"09:07.370","Text":"then we\u0027re going to infinity."},{"Start":"09:07.370 ","End":"09:11.080","Text":"In other words, the numbers keep getting larger and larger without bound."},{"Start":"09:11.080 ","End":"09:14.825","Text":"On the other side, when x go to 0 from the left,"},{"Start":"09:14.825 ","End":"09:19.000","Text":"the values get larger and larger with no limit."},{"Start":"09:19.000 ","End":"09:25.565","Text":"In other words, this side is the minus infinity and this side it goes to plus infinity,"},{"Start":"09:25.565 ","End":"09:27.515","Text":"sorry, this is also infinity."},{"Start":"09:27.515 ","End":"09:30.260","Text":"You\u0027ll see plenty more examples like this in"},{"Start":"09:30.260 ","End":"09:33.695","Text":"the exercises following the theoretical section."},{"Start":"09:33.695 ","End":"09:36.860","Text":"As in the previous sketch with 1 over x,"},{"Start":"09:36.860 ","End":"09:40.955","Text":"the y axis itself is something called an asymptote to this function;"},{"Start":"09:40.955 ","End":"09:43.500","Text":"y equals 1 over x squared."}],"ID":8411},{"Watched":false,"Name":"Exercise 1","Duration":"2m 5s","ChapterTopicVideoID":4739,"CourseChapterTopicPlaylistID":65362,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4739.jpeg","UploadDate":"2015-05-30T14:26:00.6530000","DurationForVideoObject":"PT2M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.780","Text":"In this exercise, we have to find the limit as x goes to 0 of x squared plus 4 over x."},{"Start":"00:06.780 ","End":"00:10.650","Text":"Now, if we try substituting x is equal to 0,"},{"Start":"00:10.650 ","End":"00:15.510","Text":"what we\u0027re going to get is that the denominator here is 0,"},{"Start":"00:15.510 ","End":"00:19.875","Text":"but the numerator is 0 squared plus 4 is not 0."},{"Start":"00:19.875 ","End":"00:21.149","Text":"Now, in general,"},{"Start":"00:21.149 ","End":"00:24.615","Text":"when we have the case where we have something of the form"},{"Start":"00:24.615 ","End":"00:28.260","Text":"not 0 over something that is 0,"},{"Start":"00:28.260 ","End":"00:33.275","Text":"then we have to separate the limit into the left limit and the right limit."},{"Start":"00:33.275 ","End":"00:37.130","Text":"Otherwise, we have to figure out what is the limit as x"},{"Start":"00:37.130 ","End":"00:41.180","Text":"goes to 0 from the left of the same thing,"},{"Start":"00:41.180 ","End":"00:46.745","Text":"and also the limit as x goes to 0 from the right,"},{"Start":"00:46.745 ","End":"00:48.890","Text":"which we write it like this."},{"Start":"00:48.890 ","End":"00:53.665","Text":"X squared plus 4_over_x."},{"Start":"00:53.665 ","End":"00:55.265","Text":"Let\u0027s see what this is."},{"Start":"00:55.265 ","End":"00:58.370","Text":"If we substitute 0 minus,"},{"Start":"00:58.370 ","End":"01:02.750","Text":"it\u0027s shorthand symbol for infinitesimally small amount,"},{"Start":"01:02.750 ","End":"01:05.300","Text":"close to 0 but just negative."},{"Start":"01:05.300 ","End":"01:09.690","Text":"So 0_squared plus 4 is 4 in this expression,"},{"Start":"01:09.690 ","End":"01:13.110","Text":"but when it\u0027s on its own we have to leave it as 0 minus."},{"Start":"01:13.110 ","End":"01:16.760","Text":"Which means a tiny bit and infinitesimal amount smaller than 0,"},{"Start":"01:16.760 ","End":"01:20.240","Text":"so it\u0027s 4 over something very small and negative,"},{"Start":"01:20.240 ","End":"01:24.035","Text":"which means it\u0027s going to be very large and negative"},{"Start":"01:24.035 ","End":"01:27.905","Text":"like 4 over minus 1/1,000,000 would be 4,000,000."},{"Start":"01:27.905 ","End":"01:31.080","Text":"This is equal to minus infinity, which means,"},{"Start":"01:31.080 ","End":"01:33.320","Text":"minus a very large number,"},{"Start":"01:33.320 ","End":"01:35.950","Text":"as large as you want by making this as small as you want."},{"Start":"01:35.950 ","End":"01:37.025","Text":"On the other side,"},{"Start":"01:37.025 ","End":"01:41.000","Text":"when x goes to 0 from the right means it\u0027s slightly positive,"},{"Start":"01:41.000 ","End":"01:43.610","Text":"so 0 squared plus 4 is 4,"},{"Start":"01:43.610 ","End":"01:47.840","Text":"but here we have positive 0 or positive infinitesimal amount,"},{"Start":"01:47.840 ","End":"01:49.745","Text":"and this is very large,"},{"Start":"01:49.745 ","End":"01:51.020","Text":"but positive this time."},{"Start":"01:51.020 ","End":"01:54.375","Text":"This equals just for emphasis plus infinity."},{"Start":"01:54.375 ","End":"01:58.594","Text":"Since we have here minus infinity and here we have plus infinity,"},{"Start":"01:58.594 ","End":"02:00.844","Text":"and these 2 are not equal,"},{"Start":"02:00.844 ","End":"02:04.220","Text":"then the answer is that there is no limit."},{"Start":"02:04.220 ","End":"02:06.450","Text":"It\u0027s undefined."}],"ID":4748},{"Watched":false,"Name":"Exercise 2","Duration":"1m 37s","ChapterTopicVideoID":4740,"CourseChapterTopicPlaylistID":65362,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4740.jpeg","UploadDate":"2015-05-30T14:26:10.9500000","DurationForVideoObject":"PT1M37S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.850","Text":"In this exercise, we have to find the limit as x"},{"Start":"00:02.850 ","End":"00:06.390","Text":"goes to 2 of x minus 1 squared over x minus 2."},{"Start":"00:06.390 ","End":"00:09.120","Text":"Let\u0027s see what happens if we substitute x equals 2."},{"Start":"00:09.120 ","End":"00:11.924","Text":"In the denominator 2 minus 2 is 0."},{"Start":"00:11.924 ","End":"00:14.775","Text":"Here, 2 minus 1 is 1 squared is 1,"},{"Start":"00:14.775 ","End":"00:16.320","Text":"so we get 1 over 0."},{"Start":"00:16.320 ","End":"00:22.530","Text":"The essential thing is that we get something which is non-zero over 0."},{"Start":"00:22.530 ","End":"00:25.005","Text":"Now in the case of non-zero over 0,"},{"Start":"00:25.005 ","End":"00:27.090","Text":"what we have to do is separately compute"},{"Start":"00:27.090 ","End":"00:29.520","Text":"the limit from the right and the limit from the left."},{"Start":"00:29.520 ","End":"00:32.640","Text":"In other words, we have to compute the limit as x"},{"Start":"00:32.640 ","End":"00:36.395","Text":"goes to 2 from the right of same thing,"},{"Start":"00:36.395 ","End":"00:42.620","Text":"and the limit as x goes to 2 from the left of the same thing."},{"Start":"00:42.620 ","End":"00:45.530","Text":"Substituting 2 plus we get,"},{"Start":"00:45.530 ","End":"00:48.275","Text":"2 minus 1 squared is 4,"},{"Start":"00:48.275 ","End":"00:55.480","Text":"2 plus minus 2 is what we call 0 plus a symbol for infinitesimally small but positive,"},{"Start":"00:55.480 ","End":"00:58.790","Text":"and this is equal to plus infinity."},{"Start":"00:58.790 ","End":"01:01.850","Text":"Whereas here we get 4 over"},{"Start":"01:01.850 ","End":"01:06.860","Text":"infinitesimally small but negative and that\u0027s equal to minus infinity."},{"Start":"01:06.860 ","End":"01:14.915","Text":"There is general formula that says that a/0 plus is equal to plus infinity."},{"Start":"01:14.915 ","End":"01:19.040","Text":"All this is in the case where if a is a positive quantity,"},{"Start":"01:19.040 ","End":"01:23.280","Text":"and a/0 minus is minus infinity."},{"Start":"01:23.280 ","End":"01:24.305","Text":"I\u0027ve used that here."},{"Start":"01:24.305 ","End":"01:26.600","Text":"The point to note is the right limit,"},{"Start":"01:26.600 ","End":"01:28.115","Text":"the left limit are not equal."},{"Start":"01:28.115 ","End":"01:30.785","Text":"These two things are not equal,"},{"Start":"01:30.785 ","End":"01:33.275","Text":"and when the right limit and the left limit are not equal,"},{"Start":"01:33.275 ","End":"01:35.480","Text":"then this limit doesn\u0027t exist."},{"Start":"01:35.480 ","End":"01:38.130","Text":"There is no limit."}],"ID":4749},{"Watched":false,"Name":"Exercise 3","Duration":"2m 13s","ChapterTopicVideoID":4741,"CourseChapterTopicPlaylistID":65362,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4741.jpeg","UploadDate":"2015-05-30T14:26:24.3970000","DurationForVideoObject":"PT2M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.300","Text":"In this exercise, we have to find the limit as x tends to"},{"Start":"00:03.300 ","End":"00:07.440","Text":"2 of x squared minus 1 over x minus 2, x minus 5."},{"Start":"00:07.440 ","End":"00:11.790","Text":"The first thing to try is to substitute x equals 2 and see if there\u0027s a problem."},{"Start":"00:11.790 ","End":"00:13.845","Text":"What we get on the denominator,"},{"Start":"00:13.845 ","End":"00:15.600","Text":"if x goes to 2, this is 0,"},{"Start":"00:15.600 ","End":"00:18.090","Text":"so the denominator 0, but the numerator,"},{"Start":"00:18.090 ","End":"00:21.435","Text":"2 squared minus 1 is 3 is not 0."},{"Start":"00:21.435 ","End":"00:28.274","Text":"What this is, this is something of the form non-zero over something which is 0."},{"Start":"00:28.274 ","End":"00:32.115","Text":"Or I could write it as not 0/0."},{"Start":"00:32.115 ","End":"00:34.890","Text":"In this case, what we have to do is compute"},{"Start":"00:34.890 ","End":"00:38.480","Text":"separately the left limit and the right limit and if they\u0027re equal,"},{"Start":"00:38.480 ","End":"00:40.640","Text":"then we have a limit and otherwise not."},{"Start":"00:40.640 ","End":"00:47.540","Text":"Let\u0027s look at the limit as x goes to 2 from the right of the same thing."},{"Start":"00:47.540 ","End":"00:54.340","Text":"We\u0027re going to separately look at the limit as x goes to 2 from the left."},{"Start":"00:54.340 ","End":"00:55.545","Text":"Now, in this case,"},{"Start":"00:55.545 ","End":"00:57.930","Text":"if x is 2 from the right,"},{"Start":"00:57.930 ","End":"01:01.530","Text":"then we get 2 plus minus 2 is 0 plus"},{"Start":"01:01.530 ","End":"01:05.190","Text":"which mean something infinitesimally small but positive,"},{"Start":"01:05.190 ","End":"01:07.390","Text":"so this is 0 plus,"},{"Start":"01:07.390 ","End":"01:10.610","Text":"2 minus 5 is minus 3."},{"Start":"01:10.610 ","End":"01:13.880","Text":"We don\u0027t have to write pluses and minuses except when it\u0027s 0."},{"Start":"01:13.880 ","End":"01:18.320","Text":"On the numerator, 2 squared minus 1 is 3."},{"Start":"01:18.320 ","End":"01:25.940","Text":"What we get is the denominator 0 plus times a negative quantity is 0 minus,"},{"Start":"01:25.940 ","End":"01:28.475","Text":"the one of those rules of these infinitesimals,"},{"Start":"01:28.475 ","End":"01:29.900","Text":"something very, very tiny,"},{"Start":"01:29.900 ","End":"01:32.660","Text":"but positive times a negative will be very tiny,"},{"Start":"01:32.660 ","End":"01:35.750","Text":"even negative, 0 minus, this is 3."},{"Start":"01:35.750 ","End":"01:40.145","Text":"The positive over 0 minus is minus infinity."},{"Start":"01:40.145 ","End":"01:43.415","Text":"Here we get something very similar, 3,"},{"Start":"01:43.415 ","End":"01:50.300","Text":"only when x goes to 2 minus 2 minus less 2 is something close to 0 but slightly negative."},{"Start":"01:50.300 ","End":"01:52.670","Text":"The same thing here as minus 3,"},{"Start":"01:52.670 ","End":"01:58.595","Text":"as negative 0 times a negative number equals slightly positive 0."},{"Start":"01:58.595 ","End":"01:59.735","Text":"Again, the 3."},{"Start":"01:59.735 ","End":"02:03.170","Text":"Positive over 0 plus, is plus infinity."},{"Start":"02:03.170 ","End":"02:06.335","Text":"The thing is that these 2 are not equal,"},{"Start":"02:06.335 ","End":"02:07.400","Text":"and if they\u0027re not equal,"},{"Start":"02:07.400 ","End":"02:10.385","Text":"then this thing doesn\u0027t have a limit, it doesn\u0027t exist."},{"Start":"02:10.385 ","End":"02:11.780","Text":"No limit."},{"Start":"02:11.780 ","End":"02:13.950","Text":"That is the answer."}],"ID":4750},{"Watched":false,"Name":"Exercise 4","Duration":"1m 21s","ChapterTopicVideoID":4742,"CourseChapterTopicPlaylistID":65362,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4742.jpeg","UploadDate":"2015-05-30T14:26:30.2000000","DurationForVideoObject":"PT1M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.750","Text":"In this exercise, we want to find the limit as x goes to 0"},{"Start":"00:03.750 ","End":"00:07.715","Text":"from the right of natural log of x/x."},{"Start":"00:07.715 ","End":"00:10.895","Text":"The first thing we try to do is to substitute."},{"Start":"00:10.895 ","End":"00:14.235","Text":"If we substitute x goes to 0 from the right,"},{"Start":"00:14.235 ","End":"00:18.260","Text":"then x is just 0 from the right, 0 plus."},{"Start":"00:18.260 ","End":"00:22.505","Text":"But what happens when x goes to 0 from the right of the natural logarithm?"},{"Start":"00:22.505 ","End":"00:25.470","Text":"This is actually equal to minus infinity,"},{"Start":"00:25.470 ","End":"00:28.980","Text":"and the way I usually remember this is just from the graph,"},{"Start":"00:28.980 ","End":"00:30.450","Text":"the natural logarithm,"},{"Start":"00:30.450 ","End":"00:33.030","Text":"it\u0027s only defined for x is bigger than 0,"},{"Start":"00:33.030 ","End":"00:35.490","Text":"so x could only go from the right and x is 1,"},{"Start":"00:35.490 ","End":"00:37.290","Text":"it\u0027s equal to 0,"},{"Start":"00:37.290 ","End":"00:40.080","Text":"and then it goes down asymptotically."},{"Start":"00:40.080 ","End":"00:41.565","Text":"In this area here,"},{"Start":"00:41.565 ","End":"00:43.805","Text":"when x goes to 0,"},{"Start":"00:43.805 ","End":"00:47.465","Text":"y goes down all the way to minus infinity."},{"Start":"00:47.465 ","End":"00:53.705","Text":"What we would say is the natural log of 0 plus is minus infinity."},{"Start":"00:53.705 ","End":"00:58.595","Text":"Back to here, this minus infinity over the 0 plus,"},{"Start":"00:58.595 ","End":"01:05.315","Text":"I could write it as minus infinity times 1/0 plus."},{"Start":"01:05.315 ","End":"01:08.880","Text":"Now a positive number over 0 plus is plus infinity,"},{"Start":"01:08.880 ","End":"01:12.815","Text":"so we have minus infinity times plus infinity."},{"Start":"01:12.815 ","End":"01:14.450","Text":"The minus and the plus give us"},{"Start":"01:14.450 ","End":"01:22.410","Text":"a minus and infinity times infinity is just infinity so the answer is minus infinity."}],"ID":4751},{"Watched":false,"Name":"Exercise 5","Duration":"1m 47s","ChapterTopicVideoID":4816,"CourseChapterTopicPlaylistID":65362,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4816.jpeg","UploadDate":"2015-07-15T08:03:27.1570000","DurationForVideoObject":"PT1M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.430","Text":"In this exercise, we have to find the limit of x goes to 2_minus."},{"Start":"00:04.430 ","End":"00:05.460","Text":"Now, what does this mean?"},{"Start":"00:05.460 ","End":"00:07.935","Text":"This means that x goes to 2 from the left,"},{"Start":"00:07.935 ","End":"00:12.820","Text":"the left limit of minus 1/2 natural log of 2 minus x ."},{"Start":"00:12.820 ","End":"00:14.699","Text":"In this kind of exercise,"},{"Start":"00:14.699 ","End":"00:18.600","Text":"what we do is we just substitute x is equal to 2_minus,"},{"Start":"00:18.600 ","End":"00:19.965","Text":"which is something symbolic,"},{"Start":"00:19.965 ","End":"00:27.060","Text":"but actually works because 2 less 2_minus is 0_plus,"},{"Start":"00:27.060 ","End":"00:29.955","Text":"and if this confuses you can just think at the side and say,"},{"Start":"00:29.955 ","End":"00:36.150","Text":"something like 2.000 less 1.999 and so"},{"Start":"00:36.150 ","End":"00:43.175","Text":"on and this is going to equal something like 0.001 but it could be many 0s."},{"Start":"00:43.175 ","End":"00:47.900","Text":"Something very small but positive and that\u0027s where we get this equation from."},{"Start":"00:47.900 ","End":"00:51.050","Text":"Now, if we substitute the 2 minus for x,"},{"Start":"00:51.050 ","End":"00:58.520","Text":"we get that this thing is just minus 1/2 natural log of 0_plus."},{"Start":"00:58.520 ","End":"01:02.675","Text":"Now, this natural log of 0 plus is also something you should remember,"},{"Start":"01:02.675 ","End":"01:04.040","Text":"but I\u0027m going to remind you,"},{"Start":"01:04.040 ","End":"01:05.690","Text":"this is minus infinity,"},{"Start":"01:05.690 ","End":"01:10.130","Text":"this is minus 1/2 times minus infinity and I\u0027ll just"},{"Start":"01:10.130 ","End":"01:15.115","Text":"show you quickly why this is so because if we just take a quick graph here,"},{"Start":"01:15.115 ","End":"01:16.285","Text":"1 at 0,"},{"Start":"01:16.285 ","End":"01:19.490","Text":"and this is the x-axis of course, and the y-axis."},{"Start":"01:19.490 ","End":"01:22.280","Text":"When x goes to 0 from the right,"},{"Start":"01:22.280 ","End":"01:23.720","Text":"that\u0027s what the 0_plus means,"},{"Start":"01:23.720 ","End":"01:26.015","Text":"it\u0027s like it\u0027s going here and here and here,"},{"Start":"01:26.015 ","End":"01:30.005","Text":"the function goes lower and lower and lower down to minus infinity."},{"Start":"01:30.005 ","End":"01:31.660","Text":"That\u0027s where we get this from."},{"Start":"01:31.660 ","End":"01:35.240","Text":"Now, this is equal to minus times minus is plus,"},{"Start":"01:35.240 ","End":"01:37.910","Text":"so it\u0027s 1/2 times infinity,"},{"Start":"01:37.910 ","End":"01:44.510","Text":"and anything positive times infinity is just equal to infinity because it was negative,"},{"Start":"01:44.510 ","End":"01:48.250","Text":"it would be minus infinity and that\u0027s our answer."}],"ID":4816},{"Watched":false,"Name":"Exercise 6","Duration":"2m 25s","ChapterTopicVideoID":4817,"CourseChapterTopicPlaylistID":65362,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4817.jpeg","UploadDate":"2015-07-15T08:03:40.1170000","DurationForVideoObject":"PT2M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.780","Text":"In this exercise, we have to find the limit as x goes to 0 plus."},{"Start":"00:04.780 ","End":"00:07.805","Text":"Remember that means x goes to 0 from the right,"},{"Start":"00:07.805 ","End":"00:09.650","Text":"0 plus is just a symbol."},{"Start":"00:09.650 ","End":"00:17.040","Text":"What we do in such cases is we just substitute 0 plus as if it were an actual quantity."},{"Start":"00:17.040 ","End":"00:18.720","Text":"If we do that,"},{"Start":"00:18.720 ","End":"00:20.565","Text":"what we\u0027re going to get here,"},{"Start":"00:20.565 ","End":"00:23.475","Text":"and I have to remind you of a formula that"},{"Start":"00:23.475 ","End":"00:30.405","Text":"the natural log of 0 plus is equal to minus infinity."},{"Start":"00:30.405 ","End":"00:35.680","Text":"Here we would get natural log of 0 plus is minus infinity,"},{"Start":"00:35.680 ","End":"00:44.420","Text":"we would get minus infinity squared plus twice minus infinity minus 3."},{"Start":"00:44.420 ","End":"00:47.990","Text":"Now, this thing is infinity plus infinity,"},{"Start":"00:47.990 ","End":"00:53.930","Text":"twice minus infinity is minus infinity minus 3."},{"Start":"00:53.930 ","End":"00:56.480","Text":"The problem is with this thing,"},{"Start":"00:56.480 ","End":"01:00.530","Text":"infinity minus infinity is 1 of those forms which are called indeterminate,"},{"Start":"01:00.530 ","End":"01:02.150","Text":"you can\u0027t say what it is,"},{"Start":"01:02.150 ","End":"01:05.075","Text":"in different exercises that could be different things."},{"Start":"01:05.075 ","End":"01:07.640","Text":"This approach is not going to work."},{"Start":"01:07.640 ","End":"01:08.750","Text":"What are we going to do?"},{"Start":"01:08.750 ","End":"01:11.240","Text":"Well, let\u0027s try some algebra here."},{"Start":"01:11.240 ","End":"01:14.270","Text":"What we\u0027ll do is rewrite this."},{"Start":"01:14.270 ","End":"01:18.450","Text":"We\u0027ll have the limit x goes to 0 plus."},{"Start":"01:18.450 ","End":"01:21.320","Text":"I\u0027ll take this thing and take natural log of"},{"Start":"01:21.320 ","End":"01:24.560","Text":"x outside the brackets from the first 2 terms."},{"Start":"01:24.560 ","End":"01:26.885","Text":"We\u0027ll get x goes to 0,"},{"Start":"01:26.885 ","End":"01:29.224","Text":"natural log of x,"},{"Start":"01:29.224 ","End":"01:34.670","Text":"brackets, natural log of x plus 2,"},{"Start":"01:34.670 ","End":"01:36.620","Text":"and all this minus 3."},{"Start":"01:36.620 ","End":"01:37.715","Text":"If we do this,"},{"Start":"01:37.715 ","End":"01:41.810","Text":"now we\u0027ll get better results if we put the 0 plus here,"},{"Start":"01:41.810 ","End":"01:49.935","Text":"because here we get minus infinity times minus infinity plus 2,"},{"Start":"01:49.935 ","End":"01:52.610","Text":"and all this minus 3."},{"Start":"01:52.610 ","End":"01:55.920","Text":"Now this equals minus infinity,"},{"Start":"01:55.920 ","End":"01:58.475","Text":"that whenever you have plus or minus infinity,"},{"Start":"01:58.475 ","End":"02:02.520","Text":"adding or subtracting a finite quantity doesn\u0027t change it."},{"Start":"02:02.520 ","End":"02:05.660","Text":"This is still minus infinity,"},{"Start":"02:05.660 ","End":"02:10.800","Text":"minus 3, minus infinity times minus infinity is plus infinity."},{"Start":"02:10.800 ","End":"02:14.565","Text":"So we get infinity minus 3."},{"Start":"02:14.565 ","End":"02:19.684","Text":"As I said, if you subtract finite quantity from infinity or minus infinity,"},{"Start":"02:19.684 ","End":"02:21.050","Text":"it doesn\u0027t change it,"},{"Start":"02:21.050 ","End":"02:24.350","Text":"so this is just equal to infinity."},{"Start":"02:24.350 ","End":"02:26.550","Text":"That\u0027s the answer."}],"ID":4817},{"Watched":false,"Name":"Exercise 7","Duration":"2m 39s","ChapterTopicVideoID":4818,"CourseChapterTopicPlaylistID":65362,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4818.jpeg","UploadDate":"2015-07-15T08:03:54.6070000","DurationForVideoObject":"PT2M39S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.345","Text":"In this exercise, we have to find the limit as x goes to 0 of e to the power of 1 over x."},{"Start":"00:06.345 ","End":"00:09.390","Text":"This is 1 of those cases that will be just can\u0027t substitute x equals"},{"Start":"00:09.390 ","End":"00:12.840","Text":"0 because 1 over 0 is not defined."},{"Start":"00:12.840 ","End":"00:15.630","Text":"What we do in these cases is we"},{"Start":"00:15.630 ","End":"00:19.815","Text":"separate the limit into the left limit and the right limit."},{"Start":"00:19.815 ","End":"00:24.780","Text":"In other words, what I\u0027m going to have to compute is the limit as"},{"Start":"00:24.780 ","End":"00:30.810","Text":"x goes to 0 plus of e to the power of 1 over x."},{"Start":"00:30.810 ","End":"00:39.540","Text":"Also a limit as x goes to 0 from the left of e to the power of 1 over x."},{"Start":"00:39.540 ","End":"00:42.190","Text":"We\u0027ll see what each of these is equal to."},{"Start":"00:42.190 ","End":"00:47.540","Text":"I\u0027d like to give you a small review of some properties of the exponential function."},{"Start":"00:47.540 ","End":"00:53.690","Text":"I could just write this as a formula that e to the power of infinity is equal"},{"Start":"00:53.690 ","End":"00:59.870","Text":"to infinity and e to the power of minus infinity is 0,"},{"Start":"00:59.870 ","End":"01:01.800","Text":"and you can either just remember these."},{"Start":"01:01.800 ","End":"01:04.925","Text":"Or the simplest thing is what I do is I remember"},{"Start":"01:04.925 ","End":"01:08.695","Text":"the rough shape of the graph of e to the power of x."},{"Start":"01:08.695 ","End":"01:11.840","Text":"Where this is the x-axis,"},{"Start":"01:11.840 ","End":"01:13.910","Text":"this is the y-axis,"},{"Start":"01:13.910 ","End":"01:15.590","Text":"and e to the power of x,"},{"Start":"01:15.590 ","End":"01:17.980","Text":"it goes through the 0.01."},{"Start":"01:17.980 ","End":"01:24.005","Text":"But on this side it goes down to 0 asymptotically and in here it goes to infinity."},{"Start":"01:24.005 ","End":"01:26.285","Text":"Other words, in this direction,"},{"Start":"01:26.285 ","End":"01:28.345","Text":"the graph goes down to 0,"},{"Start":"01:28.345 ","End":"01:30.320","Text":"and in this direction, the graph,"},{"Start":"01:30.320 ","End":"01:32.510","Text":"meaning y goes up to infinity."},{"Start":"01:32.510 ","End":"01:35.120","Text":"When x goes to infinity,"},{"Start":"01:35.120 ","End":"01:36.814","Text":"e to the infinity is infinity,"},{"Start":"01:36.814 ","End":"01:39.020","Text":"and when x goes to minus infinity,"},{"Start":"01:39.020 ","End":"01:41.465","Text":"e to minus infinity is 0."},{"Start":"01:41.465 ","End":"01:43.535","Text":"Let\u0027s see how we use that here."},{"Start":"01:43.535 ","End":"01:48.515","Text":"If x goes to 0 plus then e to the 1 over x,"},{"Start":"01:48.515 ","End":"01:53.805","Text":"we get e to the power of 1 over 0 plus."},{"Start":"01:53.805 ","End":"01:57.280","Text":"Now 1 over 0 plus is plus infinity."},{"Start":"01:57.280 ","End":"02:00.165","Text":"This is e to the power of infinity,"},{"Start":"02:00.165 ","End":"02:01.235","Text":"and as we said here,"},{"Start":"02:01.235 ","End":"02:02.960","Text":"this is equal to infinity."},{"Start":"02:02.960 ","End":"02:05.750","Text":"On the other hand, if x goes to 0 from the left,"},{"Start":"02:05.750 ","End":"02:09.830","Text":"then here we get e to the power of 1 over 0 minus,"},{"Start":"02:09.830 ","End":"02:12.415","Text":"which is close to 0 but negative."},{"Start":"02:12.415 ","End":"02:16.880","Text":"That\u0027s 1 over 0 minus is known to be minus infinity."},{"Start":"02:16.880 ","End":"02:20.255","Text":"Here we get e to the minus infinity,"},{"Start":"02:20.255 ","End":"02:22.960","Text":"and as we said here that\u0027s equal to 0."},{"Start":"02:22.960 ","End":"02:25.070","Text":"The thing is we took the right limit and"},{"Start":"02:25.070 ","End":"02:27.995","Text":"the left limit and they turned out to be different."},{"Start":"02:27.995 ","End":"02:30.585","Text":"This and this were not equal,"},{"Start":"02:30.585 ","End":"02:33.560","Text":"and when the left limit does not equal the right limit,"},{"Start":"02:33.560 ","End":"02:40.220","Text":"then there is no limit or limit doesn\u0027t exist or something like that, and that\u0027s it."}],"ID":4818},{"Watched":false,"Name":"Exercise 8","Duration":"1m 19s","ChapterTopicVideoID":4819,"CourseChapterTopicPlaylistID":65362,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4819.jpeg","UploadDate":"2015-07-15T13:10:31.5470000","DurationForVideoObject":"PT1M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.530","Text":"In this exercise, we have to find the limit as x goes to 0 plus,"},{"Start":"00:04.530 ","End":"00:08.190","Text":"which means x goes to 0 from the right of this expression,"},{"Start":"00:08.190 ","End":"00:10.905","Text":"1 over 1 plus 2^1 over x."},{"Start":"00:10.905 ","End":"00:14.115","Text":"This is a simple case of substitution."},{"Start":"00:14.115 ","End":"00:17.280","Text":"We substitute the symbol 0 plus,"},{"Start":"00:17.280 ","End":"00:26.280","Text":"this equals 1 over 1 plus 2^1 over 0 plus."},{"Start":"00:26.280 ","End":"00:33.000","Text":"Now, this equals 1 over 1 plus 2 to the power of,"},{"Start":"00:33.000 ","End":"00:34.545","Text":"now 1 over 0 plus,"},{"Start":"00:34.545 ","End":"00:37.200","Text":"you should remember that this equals plus infinity."},{"Start":"00:37.200 ","End":"00:38.600","Text":"Just remember, if it\u0027s a very,"},{"Start":"00:38.600 ","End":"00:40.380","Text":"very small positive number,"},{"Start":"00:40.380 ","End":"00:42.770","Text":"1 over it, the reciprocal would be a very,"},{"Start":"00:42.770 ","End":"00:44.580","Text":"very large positive number."},{"Start":"00:44.580 ","End":"00:46.575","Text":"In general, this is infinity."},{"Start":"00:46.575 ","End":"00:50.445","Text":"Now, 2^infinity is infinity,"},{"Start":"00:50.445 ","End":"00:56.150","Text":"so that\u0027s equal to 1 over 1 plus infinity."},{"Start":"00:56.150 ","End":"00:57.530","Text":"Continue on the next line,"},{"Start":"00:57.530 ","End":"01:00.230","Text":"and that\u0027s equal to 1 over."},{"Start":"01:00.230 ","End":"01:03.080","Text":"Remember, infinity or minus infinity,"},{"Start":"01:03.080 ","End":"01:07.250","Text":"if you add or subtract a finite quantity, it stays the same,"},{"Start":"01:07.250 ","End":"01:09.665","Text":"so 1 plus infinity is just infinity,"},{"Start":"01:09.665 ","End":"01:14.735","Text":"and 1 over infinity is equal to 0 plus, well, just 0."},{"Start":"01:14.735 ","End":"01:18.035","Text":"1 divided by an enormous quantity gives us 0,"},{"Start":"01:18.035 ","End":"01:20.610","Text":"and 0 is our answer."}],"ID":4819},{"Watched":false,"Name":"Exercise 9","Duration":"1m 25s","ChapterTopicVideoID":4743,"CourseChapterTopicPlaylistID":65362,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4743.jpeg","UploadDate":"2015-05-30T14:26:35.4570000","DurationForVideoObject":"PT1M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.260","Text":"In this exercise, we have to find the limit as x goes to 0 minus,"},{"Start":"00:04.260 ","End":"00:09.915","Text":"which means x goes to 0 from the left limit of 1 over 1 plus 2^1 over x."},{"Start":"00:09.915 ","End":"00:14.895","Text":"We can do this simply by substituting a set of x to 0 minus."},{"Start":"00:14.895 ","End":"00:17.250","Text":"It\u0027s very similar to the previous exercise where"},{"Start":"00:17.250 ","End":"00:19.740","Text":"we had a 0 plus here, but there\u0027s a difference."},{"Start":"00:19.740 ","End":"00:27.570","Text":"This equals 1 over 1 plus 2^1 over 0 minus,"},{"Start":"00:27.570 ","End":"00:34.980","Text":"and this equals 1 over 1 plus 2^1 over 0 minus is minus infinity,"},{"Start":"00:34.980 ","End":"00:37.730","Text":"1 over a very, very small but negative quantity is"},{"Start":"00:37.730 ","End":"00:41.610","Text":"a very large negative quantity and this gives us minus infinity."},{"Start":"00:41.610 ","End":"00:43.650","Text":"Let\u0027s see if we can figure out the side,"},{"Start":"00:43.650 ","End":"00:45.780","Text":"what is 2^minus infinity?"},{"Start":"00:45.780 ","End":"00:50.614","Text":"2^minus infinity is equal to because the rules of exponents,"},{"Start":"00:50.614 ","End":"00:54.140","Text":"1 over 2^infinity and we mentioned before"},{"Start":"00:54.140 ","End":"00:58.670","Text":"that 2^infinity is infinity just like e^infinity."},{"Start":"00:58.670 ","End":"01:06.225","Text":"In fact, what you should know is that a^infinity is infinity if a is bigger than 1."},{"Start":"01:06.225 ","End":"01:07.675","Text":"Any number larger than 1,"},{"Start":"01:07.675 ","End":"01:11.630","Text":"you keep multiplying by itself goes larger and larger and 1 over infinity,"},{"Start":"01:11.630 ","End":"01:13.400","Text":"of course, is 0."},{"Start":"01:13.400 ","End":"01:17.510","Text":"I can put this 2 to the minus infinity equals 0 back here and what"},{"Start":"01:17.510 ","End":"01:22.205","Text":"I\u0027ll get is equal to 1 over 1 plus 0,"},{"Start":"01:22.205 ","End":"01:26.190","Text":"which is just equal to 1. That\u0027s the answer."}],"ID":4752},{"Watched":false,"Name":"Exercise 10","Duration":"1m 5s","ChapterTopicVideoID":4744,"CourseChapterTopicPlaylistID":65362,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4744.jpeg","UploadDate":"2015-05-30T14:26:39.5430000","DurationForVideoObject":"PT1M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.000","Text":"In this exercise, we have to find the limit as x goes to"},{"Start":"00:03.000 ","End":"00:06.630","Text":"0 of 1 over 1 plus 2 to the 1 over x."},{"Start":"00:06.630 ","End":"00:10.650","Text":"Now I\u0027ve highlighted the 1 over x because this 1 over x fits"},{"Start":"00:10.650 ","End":"00:16.665","Text":"the definition when x goes to 0 of something which is not equal to 0,"},{"Start":"00:16.665 ","End":"00:19.455","Text":"non-zero over something which is 0."},{"Start":"00:19.455 ","End":"00:21.150","Text":"When we see this kind of situation,"},{"Start":"00:21.150 ","End":"00:24.960","Text":"we have to separate the limits into the limit from the left and limit from the right."},{"Start":"00:24.960 ","End":"00:27.540","Text":"In other words, we have to compute 2 limits."},{"Start":"00:27.540 ","End":"00:32.715","Text":"1 is the limit as x goes to 0 from the right,"},{"Start":"00:32.715 ","End":"00:36.940","Text":"and the other is the limit as x goes to 0 from the left."},{"Start":"00:36.940 ","End":"00:42.500","Text":"Now, these 2 limits are exactly the previous 2 exercises and if you haven\u0027t done them,"},{"Start":"00:42.500 ","End":"00:45.080","Text":"I suggest you take a pause and review them"},{"Start":"00:45.080 ","End":"00:47.990","Text":"because this is exactly the exercise before last."},{"Start":"00:47.990 ","End":"00:52.144","Text":"We got the answer here to equal 0 and in this exercise,"},{"Start":"00:52.144 ","End":"00:53.585","Text":"the answer was equal to 1."},{"Start":"00:53.585 ","End":"00:56.990","Text":"The point now is that the right limit and the left limit are not equal."},{"Start":"00:56.990 ","End":"00:58.610","Text":"This and this, if I compare them,"},{"Start":"00:58.610 ","End":"01:00.650","Text":"they are not equal and whenever"},{"Start":"01:00.650 ","End":"01:04.225","Text":"the left limit is not equal to the right limit then there is no limit,"},{"Start":"01:04.225 ","End":"01:06.540","Text":"so there is no limit."}],"ID":4753},{"Watched":false,"Name":"Exercise 11","Duration":"1m 25s","ChapterTopicVideoID":4745,"CourseChapterTopicPlaylistID":65362,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4745.jpeg","UploadDate":"2015-05-30T14:26:44.7400000","DurationForVideoObject":"PT1M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.080","Text":"In this exercise, we have to find the limit as x goes to 0 from the right or"},{"Start":"00:04.080 ","End":"00:08.820","Text":"x goes to 0 plus of natural log of x times the cotangent of x."},{"Start":"00:08.820 ","End":"00:13.800","Text":"All we have to do is substitute 0 plus here and here."},{"Start":"00:13.800 ","End":"00:17.495","Text":"What we get is we get the natural log."},{"Start":"00:17.495 ","End":"00:26.630","Text":"This equals the natural log of x 0 plus times the cotangent of 0 plus."},{"Start":"00:26.630 ","End":"00:28.880","Text":"Let\u0027s see. Now this,"},{"Start":"00:28.880 ","End":"00:36.050","Text":"we\u0027ve seen already many times the natural log of 0 plus is minus infinity times."},{"Start":"00:36.050 ","End":"00:39.550","Text":"Let\u0027s do this as an exercise at the side."},{"Start":"00:39.550 ","End":"00:42.560","Text":"You can remember it, but if you don\u0027t,"},{"Start":"00:42.560 ","End":"00:46.745","Text":"we can just write it as cotangent is cosine over sine."},{"Start":"00:46.745 ","End":"00:54.080","Text":"Cosine of 0 plus sine of 0 plus now the cosine of"},{"Start":"00:54.080 ","End":"01:01.915","Text":"0 is 1 and the sine of 0 plus is just 0 plus."},{"Start":"01:01.915 ","End":"01:04.100","Text":"The sine of something very, very small,"},{"Start":"01:04.100 ","End":"01:07.700","Text":"but close to 0 is also very small and positive."},{"Start":"01:07.700 ","End":"01:11.560","Text":"This thing is equal to plus infinity,"},{"Start":"01:11.560 ","End":"01:15.695","Text":"1 over 0 plus is plus infinity and so I can"},{"Start":"01:15.695 ","End":"01:20.015","Text":"write this as minus infinity times plus infinity,"},{"Start":"01:20.015 ","End":"01:26.090","Text":"which gives me minus infinity and that\u0027s the answer."}],"ID":4754}],"Thumbnail":null,"ID":65362},{"Name":"Technique 5 - X Tends to Infinity","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"X Tends to Infinity Part 1","Duration":"10m 23s","ChapterTopicVideoID":9303,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9303.jpeg","UploadDate":"2019-10-29T04:23:51.1800000","DurationForVideoObject":"PT10M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.705","Text":"In this clip, we talk about technique number 5 for limits."},{"Start":"00:03.705 ","End":"00:07.965","Text":"What to do when x tends to infinity and we write"},{"Start":"00:07.965 ","End":"00:13.110","Text":"this for a function f as limit as x tends to infinity of f of x."},{"Start":"00:13.110 ","End":"00:17.850","Text":"But of course there could also be limit as x tends to minus infinity of f of x."},{"Start":"00:17.850 ","End":"00:20.670","Text":"There is no such number as infinity,"},{"Start":"00:20.670 ","End":"00:23.550","Text":"but infinity in some ways behaves like a number."},{"Start":"00:23.550 ","End":"00:26.400","Text":"You can think of it as a giant, huge,"},{"Start":"00:26.400 ","End":"00:30.660","Text":"enormous number which keeps growing boundlessly,"},{"Start":"00:30.660 ","End":"00:32.490","Text":"no limit to the size."},{"Start":"00:32.490 ","End":"00:34.185","Text":"That\u0027s a rough idea."},{"Start":"00:34.185 ","End":"00:37.255","Text":"Nevertheless, it behaves in many ways like a number and there"},{"Start":"00:37.255 ","End":"00:40.469","Text":"even some rules which I\u0027d like to call the arithmetic of infinity."},{"Start":"00:40.469 ","End":"00:45.125","Text":"So I\u0027m going to write these rules here and call it the arithmetic of infinity."},{"Start":"00:45.125 ","End":"00:47.990","Text":"So let me start with number 1,"},{"Start":"00:47.990 ","End":"00:52.880","Text":"infinity plus infinity equals infinity,"},{"Start":"00:52.880 ","End":"00:56.120","Text":"a number which is huge and keeps growing boundlessly plus"},{"Start":"00:56.120 ","End":"01:00.145","Text":"another 1 of those is still going to be a huge number that grows without end."},{"Start":"01:00.145 ","End":"01:05.210","Text":"Number 2, infinity plus a is also equal to infinity,"},{"Start":"01:05.210 ","End":"01:08.180","Text":"where a is just an actual number something that keeps"},{"Start":"01:08.180 ","End":"01:12.110","Text":"growing endlessly and boundlessly if you add some constant to it,"},{"Start":"01:12.110 ","End":"01:17.135","Text":"like 7 or even 7 billion is still going to keep growing boundlessly."},{"Start":"01:17.135 ","End":"01:21.290","Text":"Number 3, infinity times a."},{"Start":"01:21.290 ","End":"01:22.675","Text":"This is a tricky 1,"},{"Start":"01:22.675 ","End":"01:29.345","Text":"because a could be positive or negative and this is defined piecewise if you like."},{"Start":"01:29.345 ","End":"01:31.475","Text":"So it\u0027s equal to infinity,"},{"Start":"01:31.475 ","End":"01:34.190","Text":"provided that a is a positive number."},{"Start":"01:34.190 ","End":"01:40.640","Text":"But to minus infinity if a is a negative number, what if a equals 0?"},{"Start":"01:40.640 ","End":"01:43.580","Text":"Well, that\u0027s 1 of those undefined cases"},{"Start":"01:43.580 ","End":"01:46.205","Text":"it\u0027s a meaningless we don\u0027t have infinity times 0."},{"Start":"01:46.205 ","End":"01:48.985","Text":"Next, we come to number 4,"},{"Start":"01:48.985 ","End":"01:53.810","Text":"which is infinity times infinity and again, clearly this huge,"},{"Start":"01:53.810 ","End":"01:58.610","Text":"growing boundless times another 1 of those is still going to be huge,"},{"Start":"01:58.610 ","End":"02:01.970","Text":"enormous, and boundless that\u0027s infinity."},{"Start":"02:01.970 ","End":"02:08.435","Text":"Number 5, the square root of infinity is also equal to infinity."},{"Start":"02:08.435 ","End":"02:11.735","Text":"Because if this keeps growing to a million,"},{"Start":"02:11.735 ","End":"02:13.565","Text":"a billion, a trillion,"},{"Start":"02:13.565 ","End":"02:15.920","Text":"the square root of it will grow a bit more slowly,"},{"Start":"02:15.920 ","End":"02:19.245","Text":"but it will still reach any size that you want."},{"Start":"02:19.245 ","End":"02:20.929","Text":"These 5 are basic,"},{"Start":"02:20.929 ","End":"02:24.080","Text":"there are another 3 which are going to include because they\u0027re convenient,"},{"Start":"02:24.080 ","End":"02:26.524","Text":"but they can be derived from these easily."},{"Start":"02:26.524 ","End":"02:35.585","Text":"Another 1 will be that infinity times minus infinity is equal to minus infinity."},{"Start":"02:35.585 ","End":"02:39.080","Text":"This follows because if infinity times infinity is infinity and"},{"Start":"02:39.080 ","End":"02:42.470","Text":"I put my a as minus 1 and then I use this rule,"},{"Start":"02:42.470 ","End":"02:44.194","Text":"I\u0027ll get the minus infinity."},{"Start":"02:44.194 ","End":"02:48.485","Text":"Similarly, minus infinity times minus infinity,"},{"Start":"02:48.485 ","End":"02:52.670","Text":"I can just use this rule here twice with a equals minus 1."},{"Start":"02:52.670 ","End":"02:55.115","Text":"So this time it will equal plus infinity so"},{"Start":"02:55.115 ","End":"02:58.500","Text":"the plus minuses behaved like with actual numbers."},{"Start":"02:58.500 ","End":"03:00.830","Text":"Finally, to be consistent that we have"},{"Start":"03:00.830 ","End":"03:04.490","Text":"a square root formula that\u0027s also have an infinity squared"},{"Start":"03:04.490 ","End":"03:11.240","Text":"formula and this is equal to infinity and actually this is just a rephrasing of this 1."},{"Start":"03:11.240 ","End":"03:13.565","Text":"So these are the rules of infinity."},{"Start":"03:13.565 ","End":"03:17.000","Text":"There is another 1 I forgot, number 9,"},{"Start":"03:17.000 ","End":"03:21.200","Text":"that a divided by plus or"},{"Start":"03:21.200 ","End":"03:28.550","Text":"minus infinity is equal to 0 and a can be any number plus minus or 0."},{"Start":"03:28.550 ","End":"03:31.820","Text":"There are also some undefined expressions"},{"Start":"03:31.820 ","End":"03:34.520","Text":"involving infinity and I\u0027ll give you some examples,"},{"Start":"03:34.520 ","End":"03:38.480","Text":"I\u0027ll just stress that these are undefined."},{"Start":"03:38.480 ","End":"03:45.515","Text":"For example, infinity minus infinity, 2 large, huge,"},{"Start":"03:45.515 ","End":"03:49.130","Text":"growing unbounded quantities fighting against each other,"},{"Start":"03:49.130 ","End":"03:51.290","Text":"1 pulling upwards, 1 pulling downwards,"},{"Start":"03:51.290 ","End":"03:54.410","Text":"not defined and I\u0027ll just list the rest,"},{"Start":"03:54.410 ","End":"03:58.400","Text":"I won\u0027t go into explanation, 1^infinity is undefined."},{"Start":"03:58.400 ","End":"04:02.240","Text":"Infinity over infinity is undefined,"},{"Start":"04:02.240 ","End":"04:07.190","Text":"and infinity^0 is undefined and maybe there\u0027s some more I forgot."},{"Start":"04:07.190 ","End":"04:11.750","Text":"Let\u0027s get back to the original problem that we were here to solve,"},{"Start":"04:11.750 ","End":"04:14.345","Text":"which was the limit of this form."},{"Start":"04:14.345 ","End":"04:17.570","Text":"But there\u0027s no single technique that will work for all functions"},{"Start":"04:17.570 ","End":"04:21.305","Text":"f and we\u0027re going to concentrate on 1 very common kind of f,"},{"Start":"04:21.305 ","End":"04:22.700","Text":"which is very suitable,"},{"Start":"04:22.700 ","End":"04:29.460","Text":"amenable to this technique and that is when f of x is a polynomial over a polynomial."},{"Start":"04:29.460 ","End":"04:32.450","Text":"But in case you don\u0027t know what a polynomial is, I\u0027ll spell it out."},{"Start":"04:32.450 ","End":"04:36.965","Text":"The numerator might be a times x^n."},{"Start":"04:36.965 ","End":"04:42.240","Text":"Think of it as a generalization of the quadratic ax squared plus bx plus c only"},{"Start":"04:42.240 ","End":"04:49.090","Text":"generalize ax^n plus bx^n minus 1 plus,"},{"Start":"04:49.090 ","End":"04:54.080","Text":"if there is another term may be cx^n minus 2, and so on."},{"Start":"04:54.080 ","End":"04:56.210","Text":"Until we get down to the constant."},{"Start":"04:56.210 ","End":"04:59.060","Text":"I don\u0027t know what letter of the alphabet it is so"},{"Start":"04:59.060 ","End":"05:02.195","Text":"let\u0027s just call it square box or something,"},{"Start":"05:02.195 ","End":"05:06.215","Text":"and same in the denominator also a polynomial,"},{"Start":"05:06.215 ","End":"05:07.865","Text":"let\u0027s use capital letters."},{"Start":"05:07.865 ","End":"05:14.450","Text":"A and it starts my different letter may be x^m plus"},{"Start":"05:14.450 ","End":"05:21.350","Text":"bx^m minus 1 another constant times the next power down, m minus 2."},{"Start":"05:21.350 ","End":"05:24.380","Text":"Again, I don\u0027t know how big this is and where it\u0027s going to end,"},{"Start":"05:24.380 ","End":"05:26.420","Text":"let\u0027s call it just a triangle or something."},{"Start":"05:26.420 ","End":"05:32.840","Text":"So this is called a rational function or a polynomial over a polynomial."},{"Start":"05:32.840 ","End":"05:37.650","Text":"But don\u0027t worry about the names and also if the notation confuses you,"},{"Start":"05:37.650 ","End":"05:40.250","Text":"next example will clarify all that."},{"Start":"05:40.250 ","End":"05:42.425","Text":"So let\u0027s take as an example,"},{"Start":"05:42.425 ","End":"05:50.670","Text":"the limit as x goes to infinity of 4x squared plus 10x plus 1,"},{"Start":"05:50.670 ","End":"05:54.425","Text":"there\u0027s a nice quadratic polynomial over another 1,"},{"Start":"05:54.425 ","End":"05:59.720","Text":"2x squared minus 5x plus 100."},{"Start":"05:59.720 ","End":"06:01.955","Text":"It turns out that all of these,"},{"Start":"06:01.955 ","End":"06:05.660","Text":"what I called rational function or polynomial over polynomial,"},{"Start":"06:05.660 ","End":"06:10.235","Text":"and they all fall into the category of the infinity over infinity."},{"Start":"06:10.235 ","End":"06:15.274","Text":"Now whenever we have an f of x as what I call the rational function,"},{"Start":"06:15.274 ","End":"06:18.605","Text":"which looks like this and in particular this,"},{"Start":"06:18.605 ","End":"06:22.820","Text":"there is a specific technique we use so I\u0027ll just write this out."},{"Start":"06:22.820 ","End":"06:28.040","Text":"That for this kind of exercise the technique is to take the highest power"},{"Start":"06:28.040 ","End":"06:33.320","Text":"of x outside the brackets and I\u0027ll soon explain what this means."},{"Start":"06:33.320 ","End":"06:36.965","Text":"In fact, I\u0027ll illustrate it by means of this particular exercise."},{"Start":"06:36.965 ","End":"06:40.940","Text":"The highest power of x here is x squared because the rest"},{"Start":"06:40.940 ","End":"06:44.990","Text":"is just x and this is a constant and here the highest power is x squared."},{"Start":"06:44.990 ","End":"06:48.830","Text":"So separately on the top and on the bottom we take the highest power,"},{"Start":"06:48.830 ","End":"06:52.895","Text":"which just happens to be the same in both cases, it\u0027s x squared."},{"Start":"06:52.895 ","End":"06:56.525","Text":"So this thing continuing down here,"},{"Start":"06:56.525 ","End":"07:03.350","Text":"and this is equal to the limit as x goes to infinity of x"},{"Start":"07:03.350 ","End":"07:07.280","Text":"squared and we\u0027ll take the rest outside the brackets and"},{"Start":"07:07.280 ","End":"07:11.915","Text":"likewise on the bottom we\u0027ll have x squared times something in brackets."},{"Start":"07:11.915 ","End":"07:13.760","Text":"In this case, the top,"},{"Start":"07:13.760 ","End":"07:16.265","Text":"take x squared out, you\u0027re left with 4."},{"Start":"07:16.265 ","End":"07:18.050","Text":"Take x squared out of here,"},{"Start":"07:18.050 ","End":"07:20.000","Text":"you\u0027re left with 10 over x."},{"Start":"07:20.000 ","End":"07:23.615","Text":"Here, we\u0027re left with 1 over x squared."},{"Start":"07:23.615 ","End":"07:26.000","Text":"On the bottom, take x squared out,"},{"Start":"07:26.000 ","End":"07:36.035","Text":"we\u0027re left with 2 minus 5 over x plus 100 over x squared close the brackets."},{"Start":"07:36.035 ","End":"07:37.775","Text":"Now how does this help us?"},{"Start":"07:37.775 ","End":"07:39.035","Text":"Well, this helps us."},{"Start":"07:39.035 ","End":"07:43.730","Text":"First of all, we can proceed by canceling"},{"Start":"07:43.730 ","End":"07:49.100","Text":"x squared with x squared because x is not 0 its far from 0,"},{"Start":"07:49.100 ","End":"07:52.400","Text":"it\u0027s on its way to infinity and then here everywhere else we can"},{"Start":"07:52.400 ","End":"07:56.870","Text":"substitute and using the rules that we had for the arithmetic of infinity."},{"Start":"07:56.870 ","End":"08:00.830","Text":"This comes out to be on the top we get 4,"},{"Start":"08:00.830 ","End":"08:05.120","Text":"10 over x is a over infinity that was the rule and that was"},{"Start":"08:05.120 ","End":"08:10.765","Text":"0 and also x squared is infinity and 1 over infinity is likewise 0."},{"Start":"08:10.765 ","End":"08:12.800","Text":"Following the same logic on the bottom,"},{"Start":"08:12.800 ","End":"08:16.625","Text":"we get 2 minus 0 plus 0."},{"Start":"08:16.625 ","End":"08:19.470","Text":"So altogether it\u0027s 4 over 2,"},{"Start":"08:19.470 ","End":"08:24.365","Text":"which is equal to 2 and that\u0027s the answer to this particular exercise."},{"Start":"08:24.365 ","End":"08:30.455","Text":"In general, what we do is we take the highest power of x,"},{"Start":"08:30.455 ","End":"08:37.430","Text":"which in this case would be the x^n and the highest power of x here is x^m and take"},{"Start":"08:37.430 ","End":"08:40.280","Text":"this outside the brackets and get"},{"Start":"08:40.280 ","End":"08:44.645","Text":"this thing times something in brackets and then everything else follows from there."},{"Start":"08:44.645 ","End":"08:46.700","Text":"I\u0027d like to clarify something."},{"Start":"08:46.700 ","End":"08:48.515","Text":"When I wrote what I wrote here,"},{"Start":"08:48.515 ","End":"08:51.380","Text":"I was referring to the polynomial over"},{"Start":"08:51.380 ","End":"08:56.420","Text":"polynomial case and that\u0027s what I meant by this kind of exercise."},{"Start":"08:56.420 ","End":"09:02.165","Text":"Here we take the highest power outside the brackets and usually works."},{"Start":"09:02.165 ","End":"09:05.750","Text":"But it\u0027s not restricted to just polynomial over polynomial."},{"Start":"09:05.750 ","End":"09:09.440","Text":"There are other examples where involving"},{"Start":"09:09.440 ","End":"09:14.720","Text":"polynomials where you can take the highest power of x outside the brackets and it helps."},{"Start":"09:14.720 ","End":"09:19.000","Text":"So I\u0027d just like to add an extra little clause here."},{"Start":"09:19.000 ","End":"09:21.920","Text":"Here\u0027s the extra clothes for this kind of exercise,"},{"Start":"09:21.920 ","End":"09:24.995","Text":"as well as others involving polynomials."},{"Start":"09:24.995 ","End":"09:30.740","Text":"But it does work best for the case polynomial over polynomial and in fact,"},{"Start":"09:30.740 ","End":"09:33.080","Text":"in these cases is even a shortcut,"},{"Start":"09:33.080 ","End":"09:34.925","Text":"I\u0027ll show you in a second."},{"Start":"09:34.925 ","End":"09:39.320","Text":"Notice where this 2 came from in this particular example."},{"Start":"09:39.320 ","End":"09:40.835","Text":"If we follow it back,"},{"Start":"09:40.835 ","End":"09:42.755","Text":"it came from 4 over 2,"},{"Start":"09:42.755 ","End":"09:44.870","Text":"which came from here,"},{"Start":"09:44.870 ","End":"09:50.330","Text":"which ultimately came from 4x squared over 2x squared and this is a general rule that"},{"Start":"09:50.330 ","End":"09:55.955","Text":"when you have a polynomial over polynomial and you take out just the highest,"},{"Start":"09:55.955 ","End":"09:58.910","Text":"in this case, the highest power is 4x squared."},{"Start":"09:58.910 ","End":"10:01.190","Text":"In this case, the highest power is"},{"Start":"10:01.190 ","End":"10:04.580","Text":"2x squared be careful though it needn\u0027t be the first term."},{"Start":"10:04.580 ","End":"10:07.250","Text":"The highest power could be somewhere in the middle and if we just"},{"Start":"10:07.250 ","End":"10:10.759","Text":"compute the limit of these highest powers,"},{"Start":"10:10.759 ","End":"10:13.565","Text":"1 from the top, 1 from the bottom and ignore all the rest,"},{"Start":"10:13.565 ","End":"10:15.140","Text":"then you\u0027ll get the same answer."},{"Start":"10:15.140 ","End":"10:17.660","Text":"So that\u0027s something very useful,"},{"Start":"10:17.660 ","End":"10:21.740","Text":"very practical and soon we\u0027ll see this in the exercises."},{"Start":"10:21.740 ","End":"10:24.360","Text":"We\u0027ll have some examples."}],"ID":9615},{"Watched":false,"Name":"X Tends to Infinity Part 2","Duration":"7m 40s","ChapterTopicVideoID":9304,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/9304.jpeg","UploadDate":"2019-10-29T04:22:10.4030000","DurationForVideoObject":"PT7M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.490","Text":"Let\u0027s proceed with the examples."},{"Start":"00:02.490 ","End":"00:05.700","Text":"All of these examples on this page will be polynomials"},{"Start":"00:05.700 ","End":"00:08.970","Text":"over polynomials and we\u0027ll practice that shortcut."},{"Start":"00:08.970 ","End":"00:14.370","Text":"The first 1 is as written here and the first thing we do is to find"},{"Start":"00:14.370 ","End":"00:20.800","Text":"the highest power of x in each and very quickly we find that it\u0027s this 1, the x^4."},{"Start":"00:20.800 ","End":"00:24.765","Text":"We circled the whole term including the coefficient."},{"Start":"00:24.765 ","End":"00:32.160","Text":"On the bottom, we find that it\u0027s the 100x^4, we circle that."},{"Start":"00:32.160 ","End":"00:34.745","Text":"Then we continue to do the limit,"},{"Start":"00:34.745 ","End":"00:36.635","Text":"ignoring all the other terms."},{"Start":"00:36.635 ","End":"00:41.540","Text":"This is the limit as x goes to infinity of"},{"Start":"00:41.540 ","End":"00:48.060","Text":"10x^4, over 100x^4."},{"Start":"00:48.060 ","End":"00:52.640","Text":"Coincidentally, again we have an equal power at the top and the bottom,"},{"Start":"00:52.640 ","End":"00:56.180","Text":"x^4 and x^4, and the whole thing cancels and we\u0027re left with"},{"Start":"00:56.180 ","End":"01:00.085","Text":"just 10/100, which is 1/10."},{"Start":"01:00.085 ","End":"01:02.040","Text":"Now, the next example."},{"Start":"01:02.040 ","End":"01:09.210","Text":"Limit as x goes to infinity of 2x squared minus 5 over 3x plus 7."},{"Start":"01:09.210 ","End":"01:11.565","Text":"Circle the highest powers,"},{"Start":"01:11.565 ","End":"01:13.680","Text":"in this case, 2x squared,"},{"Start":"01:13.680 ","End":"01:16.290","Text":"in this case the 3x."},{"Start":"01:16.290 ","End":"01:19.695","Text":"Then this is equal to limit,"},{"Start":"01:19.695 ","End":"01:22.315","Text":"and we just throw out the rest of the terms,"},{"Start":"01:22.315 ","End":"01:27.600","Text":"of 2x squared over 3x."},{"Start":"01:27.600 ","End":"01:30.940","Text":"Now we do have different powers at the top and the bottom,"},{"Start":"01:30.940 ","End":"01:32.255","Text":"the top is higher."},{"Start":"01:32.255 ","End":"01:34.460","Text":"Still we can do a partial cancellation,"},{"Start":"01:34.460 ","End":"01:35.975","Text":"x squared over x,"},{"Start":"01:35.975 ","End":"01:39.715","Text":"this gives us just the x on the top."},{"Start":"01:39.715 ","End":"01:46.115","Text":"Then I substitute x equals infinity and we get 2/3 of infinity basically."},{"Start":"01:46.115 ","End":"01:53.095","Text":"We do the arithmetic for infinity and positive number times infinity is infinity."},{"Start":"01:53.095 ","End":"01:56.845","Text":"That\u0027s this 1. Now on to the next 1."},{"Start":"01:56.845 ","End":"02:00.050","Text":"Limit as x goes to minus infinity,"},{"Start":"02:00.050 ","End":"02:03.680","Text":"this time of 2x squared plus 1 over 10x."},{"Start":"02:03.680 ","End":"02:07.350","Text":"We circle the highest powers,"},{"Start":"02:07.350 ","End":"02:09.135","Text":"this is the usual technique."},{"Start":"02:09.135 ","End":"02:11.780","Text":"Here it\u0027s this and here there is only 1 term,"},{"Start":"02:11.780 ","End":"02:13.675","Text":"so must be this."},{"Start":"02:13.675 ","End":"02:16.395","Text":"Then we continue, limit,"},{"Start":"02:16.395 ","End":"02:18.300","Text":"still x goes to infinity,"},{"Start":"02:18.300 ","End":"02:23.025","Text":"of 2x squared over 10x."},{"Start":"02:23.025 ","End":"02:27.170","Text":"Once again, we have a higher power in the top,"},{"Start":"02:27.170 ","End":"02:29.280","Text":"but we can partially cancel,"},{"Start":"02:29.280 ","End":"02:31.515","Text":"we even got x squared and x again."},{"Start":"02:31.515 ","End":"02:34.020","Text":"This x cancels 1 of the x\u0027s here."},{"Start":"02:34.020 ","End":"02:37.710","Text":"It\u0027s like I break through the 2 and this equals,"},{"Start":"02:37.710 ","End":"02:39.585","Text":"if I put x equals infinity,"},{"Start":"02:39.585 ","End":"02:43.970","Text":"2/10 times infinity and this is on these rules"},{"Start":"02:43.970 ","End":"02:48.470","Text":"that a positive number times infinity is equal to infinity,"},{"Start":"02:48.470 ","End":"02:50.755","Text":"and that\u0027s this 1."},{"Start":"02:50.755 ","End":"02:52.480","Text":"Let\u0027s do another 1."},{"Start":"02:52.480 ","End":"02:56.450","Text":"Limit as x goes to infinity of all this stuff."},{"Start":"02:56.450 ","End":"02:58.460","Text":"Identify the highest powers,"},{"Start":"02:58.460 ","End":"03:00.725","Text":"they are the only ones that matter."},{"Start":"03:00.725 ","End":"03:04.490","Text":"In this case, it\u0027s the x^4 term,"},{"Start":"03:04.490 ","End":"03:07.165","Text":"the whole term I circle."},{"Start":"03:07.165 ","End":"03:09.510","Text":"Here that\u0027s a 5 there,"},{"Start":"03:09.510 ","End":"03:13.110","Text":"this is x^5, that\u0027s obviously the highest term, circle this."},{"Start":"03:13.110 ","End":"03:17.950","Text":"Then just write it as the limit x goes to"},{"Start":"03:17.950 ","End":"03:24.700","Text":"infinity of 2x^4 over 10x^5."},{"Start":"03:24.700 ","End":"03:28.705","Text":"In this case, the bottom has the higher power this time,"},{"Start":"03:28.705 ","End":"03:31.120","Text":"this is x^5, this is x^4."},{"Start":"03:31.120 ","End":"03:33.670","Text":"When we cancel x^4,"},{"Start":"03:33.670 ","End":"03:36.250","Text":"cancels 2x^5 x times."},{"Start":"03:36.250 ","End":"03:38.985","Text":"I can just cross out the 5."},{"Start":"03:38.985 ","End":"03:43.065","Text":"What I get is 1/5 over x,"},{"Start":"03:43.065 ","End":"03:45.960","Text":"the x is infinity like that."},{"Start":"03:45.960 ","End":"03:52.905","Text":"a over infinity for any a is equal to 0. That\u0027s this 1."},{"Start":"03:52.905 ","End":"03:56.405","Text":"Now I\u0027d like to and I\u0027ll just do it verbally I won\u0027t write it down,"},{"Start":"03:56.405 ","End":"04:01.845","Text":"we\u0027ve used the ultimate shortcut for when you have a polynomial over a polynomial."},{"Start":"04:01.845 ","End":"04:07.880","Text":"This is the rule, if the highest powers are equal x^4 and x^4,"},{"Start":"04:07.880 ","End":"04:11.665","Text":"then the answer is just the 10/100,"},{"Start":"04:11.665 ","End":"04:13.395","Text":"just take the coefficients,"},{"Start":"04:13.395 ","End":"04:16.320","Text":"10/ 100, which is 1/10."},{"Start":"04:16.320 ","End":"04:19.730","Text":"If the highest power on"},{"Start":"04:19.730 ","End":"04:24.335","Text":"the numerator is higher than the highest power on the denominator,"},{"Start":"04:24.335 ","End":"04:27.890","Text":"then the answer is going to be plus or minus infinity according to"},{"Start":"04:27.890 ","End":"04:31.860","Text":"the signs and according to whether it\u0027s x goes to plus or minus infinity,"},{"Start":"04:31.860 ","End":"04:34.675","Text":"but other than the plus or minus is going to be infinity."},{"Start":"04:34.675 ","End":"04:36.080","Text":"We had an example here,"},{"Start":"04:36.080 ","End":"04:37.640","Text":"and we had an example here."},{"Start":"04:37.640 ","End":"04:41.029","Text":"The third case is where it\u0027s higher on the bottom,"},{"Start":"04:41.029 ","End":"04:45.980","Text":"like 4 versus 5 and in that case the answer is always going to be 0."},{"Start":"04:45.980 ","End":"04:48.425","Text":"That\u0027s basically your rules."},{"Start":"04:48.425 ","End":"04:50.660","Text":"Now, I\u0027d like to go on to do"},{"Start":"04:50.660 ","End":"04:54.050","Text":"something a little bit different, slightly different example."},{"Start":"04:54.050 ","End":"04:58.295","Text":"I want to break out of this mold of polynomial over polynomial,"},{"Start":"04:58.295 ","End":"05:00.920","Text":"also known as rational function."},{"Start":"05:00.920 ","End":"05:07.500","Text":"I want to do something that isn\u0027t just like that and my example is going to be here."},{"Start":"05:07.500 ","End":"05:10.310","Text":"A different example, not the usual."},{"Start":"05:10.310 ","End":"05:14.120","Text":"Notice, this is a polynomial and this is a polynomial,"},{"Start":"05:14.120 ","End":"05:18.995","Text":"but this 1 is under a square root sign, something completely different."},{"Start":"05:18.995 ","End":"05:20.720","Text":"Now, we can\u0027t use"},{"Start":"05:20.720 ","End":"05:25.820","Text":"that shortcut trick of just taking the highest power in each polynomial,"},{"Start":"05:25.820 ","End":"05:31.930","Text":"but we still can take out the highest power as a factor that\u0027s still holds."},{"Start":"05:31.930 ","End":"05:33.900","Text":"Let\u0027s do that. Now,"},{"Start":"05:33.900 ","End":"05:37.490","Text":"the highest power here is the x squared,"},{"Start":"05:37.490 ","End":"05:40.085","Text":"and the highest power here is the x."},{"Start":"05:40.085 ","End":"05:45.785","Text":"In each case, we take the highest power outside of its corresponding polynomial."},{"Start":"05:45.785 ","End":"05:47.300","Text":"In the first case,"},{"Start":"05:47.300 ","End":"05:48.895","Text":"we\u0027ll get the limit,"},{"Start":"05:48.895 ","End":"05:53.330","Text":"x goes to infinity of the square root."},{"Start":"05:53.330 ","End":"05:56.645","Text":"Now taking x squared outside the brackets,"},{"Start":"05:56.645 ","End":"06:00.645","Text":"what we\u0027re left with is x squared out of x squared leaves 1,"},{"Start":"06:00.645 ","End":"06:04.125","Text":"out of the x leaves 1 over x,"},{"Start":"06:04.125 ","End":"06:07.085","Text":"and here, 1 over x squared."},{"Start":"06:07.085 ","End":"06:14.345","Text":"On the bottom, the x comes out and we\u0027re left with 4 plus 1 over x."},{"Start":"06:14.345 ","End":"06:21.020","Text":"Now, I can take this square root and apply it separately to each factor."},{"Start":"06:21.020 ","End":"06:22.640","Text":"I get the limit,"},{"Start":"06:22.640 ","End":"06:24.710","Text":"still x goes to infinity,"},{"Start":"06:24.710 ","End":"06:27.890","Text":"of the square root of x squared times"},{"Start":"06:27.890 ","End":"06:33.170","Text":"the square root of 1 plus 1 over x plus 1 over x squared,"},{"Start":"06:33.170 ","End":"06:36.400","Text":"and all this is over x."},{"Start":"06:36.400 ","End":"06:40.095","Text":"Here, 4 plus 1 over x."},{"Start":"06:40.095 ","End":"06:42.725","Text":"Now, what\u0027s the square root of x squared?"},{"Start":"06:42.725 ","End":"06:46.775","Text":"Everyone will say x, and it\u0027s true in this case,"},{"Start":"06:46.775 ","End":"06:48.410","Text":"but it\u0027s not always true."},{"Start":"06:48.410 ","End":"06:50.720","Text":"In general just remember that the square root of x"},{"Start":"06:50.720 ","End":"06:53.450","Text":"squared is equal to the absolute value of x."},{"Start":"06:53.450 ","End":"06:56.965","Text":"But of course here x is positive on its way to infinity,"},{"Start":"06:56.965 ","End":"06:59.235","Text":"so it\u0027s certainly positive."},{"Start":"06:59.235 ","End":"07:01.310","Text":"It makes sense that this thing is x,"},{"Start":"07:01.310 ","End":"07:05.900","Text":"in which case this thing cancels and all we\u0027re left with"},{"Start":"07:05.900 ","End":"07:11.700","Text":"is the square root of 1 plus 0 plus 0,"},{"Start":"07:11.700 ","End":"07:16.100","Text":"because all these thing is1 over infinity and 1 over infinity is 0."},{"Start":"07:16.100 ","End":"07:19.460","Text":"Also we\u0027re here 1 over infinity is 0,"},{"Start":"07:19.460 ","End":"07:25.140","Text":"so over 4 and that just leaves us with 1 quarter."},{"Start":"07:25.140 ","End":"07:30.020","Text":"This is the example that\u0027s different from the rest and"},{"Start":"07:30.020 ","End":"07:32.705","Text":"basically we\u0027re done for this clip"},{"Start":"07:32.705 ","End":"07:35.690","Text":"but there are lots of exercises and you need to practice,"},{"Start":"07:35.690 ","End":"07:40.810","Text":"so I\u0027ll leave you to get on with those. That\u0027s all."}],"ID":9616},{"Watched":false,"Name":"Exercise 1","Duration":"1m 29s","ChapterTopicVideoID":4747,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4747.jpeg","UploadDate":"2016-05-01T15:16:22.6970000","DurationForVideoObject":"PT1M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.535","Text":"In this exercise, we have to find the limit as x goes to infinity of e to the minus x,"},{"Start":"00:05.535 ","End":"00:08.310","Text":"all of these to the power of natural log of x."},{"Start":"00:08.310 ","End":"00:13.545","Text":"The usual thing to do with infinity is just to substitute it and if we substituted it,"},{"Start":"00:13.545 ","End":"00:19.455","Text":"what we would get would be e to the power of minus infinity,"},{"Start":"00:19.455 ","End":"00:24.365","Text":"to the power of natural log of infinity."},{"Start":"00:24.365 ","End":"00:27.770","Text":"Now, this equals e to the minus infinity is"},{"Start":"00:27.770 ","End":"00:31.490","Text":"0 and the natural log of infinity is infinity."},{"Start":"00:31.490 ","End":"00:33.290","Text":"We get 0 to the infinity,"},{"Start":"00:33.290 ","End":"00:35.600","Text":"which is another 1 of those indeterminate forms."},{"Start":"00:35.600 ","End":"00:36.680","Text":"We can\u0027t say what it is."},{"Start":"00:36.680 ","End":"00:39.905","Text":"It could be different things in different exercises."},{"Start":"00:39.905 ","End":"00:42.245","Text":"What we\u0027re going do is a tiny trick."},{"Start":"00:42.245 ","End":"00:44.450","Text":"If you remember the rules of exponents,"},{"Start":"00:44.450 ","End":"00:48.170","Text":"what we could do is to write this as the limit,"},{"Start":"00:48.170 ","End":"00:50.175","Text":"x goes to infinity."},{"Start":"00:50.175 ","End":"00:52.190","Text":"When we have an exponent of an exponent,"},{"Start":"00:52.190 ","End":"00:53.660","Text":"you multiply the exponents,"},{"Start":"00:53.660 ","End":"01:00.259","Text":"so we get e to the power of minus x times natural log of x."},{"Start":"01:00.259 ","End":"01:02.240","Text":"Now we do the substitution,"},{"Start":"01:02.240 ","End":"01:04.130","Text":"we get, if x is infinity,"},{"Start":"01:04.130 ","End":"01:07.835","Text":"we get e to the power of minus infinity,"},{"Start":"01:07.835 ","End":"01:09.950","Text":"and the natural log of infinity,"},{"Start":"01:09.950 ","End":"01:11.270","Text":"we\u0027ve already mentioned,"},{"Start":"01:11.270 ","End":"01:13.790","Text":"is infinity times infinity."},{"Start":"01:13.790 ","End":"01:18.440","Text":"Now infinity times infinity is infinity so we get e"},{"Start":"01:18.440 ","End":"01:23.105","Text":"to the minus infinity and e to the power of minus infinity."},{"Start":"01:23.105 ","End":"01:24.955","Text":"We\u0027ve seen that many times."},{"Start":"01:24.955 ","End":"01:29.470","Text":"that\u0027s equal to 0 and that\u0027s our answer."}],"ID":4756},{"Watched":false,"Name":"Exercise 2","Duration":"1m 33s","ChapterTopicVideoID":4748,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4748.jpeg","UploadDate":"2016-05-01T15:16:30.8030000","DurationForVideoObject":"PT1M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.480","Text":"In this exercise, we have to find the limit as x goes to"},{"Start":"00:03.480 ","End":"00:07.830","Text":"minus infinity of arctangent x plus e^x."},{"Start":"00:07.830 ","End":"00:09.375","Text":"In the case of limits,"},{"Start":"00:09.375 ","End":"00:12.405","Text":"if a plus or minus infinity, we just substitute."},{"Start":"00:12.405 ","End":"00:18.270","Text":"What we get is that this thing is equal to the arctangent"},{"Start":"00:18.270 ","End":"00:24.795","Text":"of minus infinity plus e to the power of minus infinity."},{"Start":"00:24.795 ","End":"00:28.920","Text":"The arctangent of minus infinity is known from trigonometry."},{"Start":"00:28.920 ","End":"00:35.310","Text":"From before you\u0027ve seen this is minus Pi over 2 and e to the minus infinity."},{"Start":"00:35.310 ","End":"00:38.610","Text":"We\u0027ve also seen before that this is equal to"},{"Start":"00:38.610 ","End":"00:43.280","Text":"0 and the answer is minus Pi over 2, and we\u0027re done."},{"Start":"00:43.280 ","End":"00:45.620","Text":"But for those who want a bit of further explanation,"},{"Start":"00:45.620 ","End":"00:52.775","Text":"the arctangent function is something like an asymptote at plus and minus Pi over 2."},{"Start":"00:52.775 ","End":"00:55.220","Text":"This is Pi over 2,"},{"Start":"00:55.220 ","End":"00:58.920","Text":"this is minus Pi over 2."},{"Start":"00:58.920 ","End":"01:02.760","Text":"The function basically looks like this."},{"Start":"01:02.760 ","End":"01:05.405","Text":"When something goes to minus infinity,"},{"Start":"01:05.405 ","End":"01:08.675","Text":"the arctangent goes to minus Pi over 2."},{"Start":"01:08.675 ","End":"01:10.130","Text":"As for this, again,"},{"Start":"01:10.130 ","End":"01:16.130","Text":"we can look at the graph and e^x goes something like goes through 0,"},{"Start":"01:16.130 ","End":"01:18.475","Text":"1, something like this."},{"Start":"01:18.475 ","End":"01:21.380","Text":"Here it goes to 0, the y,"},{"Start":"01:21.380 ","End":"01:23.720","Text":"and here it goes to infinity."},{"Start":"01:23.720 ","End":"01:26.720","Text":"It\u0027s like here it goes to Pi over 2,"},{"Start":"01:26.720 ","End":"01:29.690","Text":"and here it goes to minus Pi over 2."},{"Start":"01:29.690 ","End":"01:34.440","Text":"So quick look at the graphs will tell you. We\u0027re done."}],"ID":4757},{"Watched":false,"Name":"Exercise 3","Duration":"2m 9s","ChapterTopicVideoID":4749,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4749.jpeg","UploadDate":"2016-05-01T15:16:41.6930000","DurationForVideoObject":"PT2M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.280","Text":"In this exercise, we have to find the limit as x goes to infinity of this expression,"},{"Start":"00:05.280 ","End":"00:07.530","Text":"which is a polynomial over a polynomial."},{"Start":"00:07.530 ","End":"00:08.640","Text":"When x goes to infinity,"},{"Start":"00:08.640 ","End":"00:11.870","Text":"the usual thing to do would be to substitute x equals infinity,"},{"Start":"00:11.870 ","End":"00:13.070","Text":"and if we did that,"},{"Start":"00:13.070 ","End":"00:17.940","Text":"we would get 4 times infinity squared plus 2 over"},{"Start":"00:17.940 ","End":"00:23.160","Text":"infinity squared plus 1,000 times infinity."},{"Start":"00:23.160 ","End":"00:24.540","Text":"If you computed this,"},{"Start":"00:24.540 ","End":"00:27.675","Text":"you\u0027d see that what we get is infinity over infinity,"},{"Start":"00:27.675 ","End":"00:29.220","Text":"which is an indeterminate form."},{"Start":"00:29.220 ","End":"00:30.690","Text":"We can\u0027t say what this is."},{"Start":"00:30.690 ","End":"00:33.330","Text":"So we\u0027re going to have to try another approach."},{"Start":"00:33.330 ","End":"00:35.670","Text":"Let me just erase this."},{"Start":"00:35.670 ","End":"00:38.385","Text":"This is not the way to do it."},{"Start":"00:38.385 ","End":"00:41.000","Text":"Instead, we\u0027re going to use a trick which is often"},{"Start":"00:41.000 ","End":"00:43.580","Text":"used when we have a polynomial over a polynomial,"},{"Start":"00:43.580 ","End":"00:45.320","Text":"and that trick is to take out"},{"Start":"00:45.320 ","End":"00:49.160","Text":"the highest power of x in the numerator and in the denominator."},{"Start":"00:49.160 ","End":"00:53.060","Text":"Rewriting this, we get the limit x"},{"Start":"00:53.060 ","End":"00:57.710","Text":"goes to infinity and the highest power of x here is x squared."},{"Start":"00:57.710 ","End":"00:59.360","Text":"We\u0027ll take that out,"},{"Start":"00:59.360 ","End":"01:00.905","Text":"and that\u0027s x squared."},{"Start":"01:00.905 ","End":"01:04.865","Text":"What we\u0027re left with is here we get a 4 and a 2."},{"Start":"01:04.865 ","End":"01:07.430","Text":"We have to divide by x squared,"},{"Start":"01:07.430 ","End":"01:09.485","Text":"which is what we took out the brackets."},{"Start":"01:09.485 ","End":"01:11.045","Text":"In the denominator,"},{"Start":"01:11.045 ","End":"01:13.055","Text":"the highest power is also x squared,"},{"Start":"01:13.055 ","End":"01:14.360","Text":"so it\u0027s x squared."},{"Start":"01:14.360 ","End":"01:16.310","Text":"This time, we\u0027re left with 1."},{"Start":"01:16.310 ","End":"01:19.265","Text":"If you do a bit of algebra, x over x squared is 1 over x."},{"Start":"01:19.265 ","End":"01:23.539","Text":"So we have here 1,000 over x."},{"Start":"01:23.539 ","End":"01:26.675","Text":"Now, we\u0027re lucky, this thing cancels."},{"Start":"01:26.675 ","End":"01:32.840","Text":"So what we\u0027re left with is 4 plus 2 over infinity"},{"Start":"01:32.840 ","End":"01:39.530","Text":"squared all over 1 plus 1,000 over infinity."},{"Start":"01:39.530 ","End":"01:43.250","Text":"At this point, I just like to remind you of something that in general,"},{"Start":"01:43.250 ","End":"01:44.300","Text":"if we have any number,"},{"Start":"01:44.300 ","End":"01:46.430","Text":"a finite number, not usual number,"},{"Start":"01:46.430 ","End":"01:48.535","Text":"and we divide it by infinity,"},{"Start":"01:48.535 ","End":"01:50.720","Text":"actually, it could be minus infinity."},{"Start":"01:50.720 ","End":"01:54.470","Text":"So I\u0027ll write plus or minus infinity. This equals 0."},{"Start":"01:54.470 ","End":"01:56.540","Text":"Using that fact here,"},{"Start":"01:56.540 ","End":"01:59.285","Text":"what we get is 4 plus,"},{"Start":"01:59.285 ","End":"02:00.800","Text":"and this thing is 0,"},{"Start":"02:00.800 ","End":"02:03.640","Text":"over 1 plus 0."},{"Start":"02:03.640 ","End":"02:09.520","Text":"In short, the answer is just 4, and we\u0027re done."}],"ID":4758},{"Watched":false,"Name":"Exercise 4","Duration":"2m 41s","ChapterTopicVideoID":4750,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4750.jpeg","UploadDate":"2016-05-01T15:16:55.5130000","DurationForVideoObject":"PT2M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.440","Text":"In this exercise, we have to find the limit as x goes to minus infinity."},{"Start":"00:04.440 ","End":"00:07.890","Text":"Again, we have a polynomial over a polynomial,"},{"Start":"00:07.890 ","End":"00:10.320","Text":"similar to the previous exercise."},{"Start":"00:10.320 ","End":"00:14.160","Text":"If we were to substitute x equals minus infinity here,"},{"Start":"00:14.160 ","End":"00:19.800","Text":"what we would get would be infinity over minus infinity,"},{"Start":"00:19.800 ","End":"00:22.260","Text":"which is another one of those forms that you can\u0027t"},{"Start":"00:22.260 ","End":"00:24.870","Text":"say what it is so this is not the approach."},{"Start":"00:24.870 ","End":"00:29.550","Text":"Instead, we\u0027re going to use the same trick as we did before by taking out"},{"Start":"00:29.550 ","End":"00:34.480","Text":"the highest power of x in the numerator and in the denominator. Let\u0027s do that."},{"Start":"00:34.480 ","End":"00:40.595","Text":"We have the limit as x tends to minus infinity."},{"Start":"00:40.595 ","End":"00:42.695","Text":"Now here in the numerator,"},{"Start":"00:42.695 ","End":"00:47.595","Text":"I can take out x^4 so I have x^4."},{"Start":"00:47.595 ","End":"00:52.040","Text":"What I\u0027m left with is 1 plus 2 over x"},{"Start":"00:52.040 ","End":"00:57.835","Text":"squared plus 6 over x^4 as the numerator."},{"Start":"00:57.835 ","End":"01:04.400","Text":"On the denominator, I can take out x cubed so I get x cubed times, here,"},{"Start":"01:04.400 ","End":"01:09.205","Text":"I have 3 plus 10 over x squared,"},{"Start":"01:09.205 ","End":"01:12.840","Text":"because x over x cubed is x squared."},{"Start":"01:12.840 ","End":"01:15.450","Text":"We can do a little bit of canceling,"},{"Start":"01:15.450 ","End":"01:21.625","Text":"the x cubed with the x^4 just leaves us with x so we just have x left here."},{"Start":"01:21.625 ","End":"01:26.145","Text":"Now, we can substitute minus infinity."},{"Start":"01:26.145 ","End":"01:28.400","Text":"From this x here,"},{"Start":"01:28.400 ","End":"01:34.880","Text":"we will have minus infinity times 1 plus 2 over minus infinity squared,"},{"Start":"01:34.880 ","End":"01:37.355","Text":"but minus infinity squared is plus infinity,"},{"Start":"01:37.355 ","End":"01:40.060","Text":"and also minus infinity to the power of 4,"},{"Start":"01:40.060 ","End":"01:45.360","Text":"an even power is also infinity, 6 over infinity."},{"Start":"01:45.360 ","End":"01:47.055","Text":"On the denominator,"},{"Start":"01:47.055 ","End":"01:50.235","Text":"left with 3 plus 10 over,"},{"Start":"01:50.235 ","End":"01:55.255","Text":"again, minus infinity squared is plus infinity so it\u0027s the 10 over infinity."},{"Start":"01:55.255 ","End":"01:58.280","Text":"I want to remind you that whenever we have"},{"Start":"01:58.280 ","End":"02:02.540","Text":"a regular finite number and we divide it by infinity,"},{"Start":"02:02.540 ","End":"02:04.295","Text":"could be plus or minus,"},{"Start":"02:04.295 ","End":"02:05.975","Text":"this is equal to 0."},{"Start":"02:05.975 ","End":"02:07.550","Text":"If I can apply that here,"},{"Start":"02:07.550 ","End":"02:09.070","Text":"here, and here,"},{"Start":"02:09.070 ","End":"02:14.870","Text":"what we\u0027d be left with is minus infinity times 1"},{"Start":"02:14.870 ","End":"02:21.635","Text":"plus 0 plus 0 over 3 plus 0,"},{"Start":"02:21.635 ","End":"02:27.720","Text":"which equals minus infinity times 1/3."},{"Start":"02:27.720 ","End":"02:30.030","Text":"This is equal to,"},{"Start":"02:30.030 ","End":"02:34.505","Text":"when we multiply minus infinity by positive quantity,"},{"Start":"02:34.505 ","End":"02:36.770","Text":"we\u0027re still left with minus infinity,"},{"Start":"02:36.770 ","End":"02:41.550","Text":"so it\u0027s just minus infinity. That\u0027s the answer."}],"ID":4759},{"Watched":false,"Name":"Exercise 5","Duration":"2m 33s","ChapterTopicVideoID":4751,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4751.jpeg","UploadDate":"2016-05-01T15:17:08.4270000","DurationForVideoObject":"PT2M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"In this exercise, we have to find the limit as x goes to"},{"Start":"00:03.270 ","End":"00:06.990","Text":"infinity of this expression polynomial over polynomial,"},{"Start":"00:06.990 ","End":"00:08.370","Text":"seen this thing before."},{"Start":"00:08.370 ","End":"00:13.035","Text":"The usual thing to do is to try and substitute x equals infinity,"},{"Start":"00:13.035 ","End":"00:20.070","Text":"and very quickly see that this comes out to be of the form infinity over infinity,"},{"Start":"00:20.070 ","End":"00:22.995","Text":"which is 1 of those undefined indeterminate forms."},{"Start":"00:22.995 ","End":"00:26.850","Text":"Which means that we have to use our favorite trick of"},{"Start":"00:26.850 ","End":"00:31.080","Text":"taking out the factor out of the numerator and the denominator."},{"Start":"00:31.080 ","End":"00:37.815","Text":"In this case, we get the limit as x goes to infinity."},{"Start":"00:37.815 ","End":"00:44.020","Text":"Here the highest power is x_4 so we take x_4 out of the brackets,"},{"Start":"00:44.020 ","End":"00:53.140","Text":"and we\u0027re left with 1 plus 2 over x squared plus 6 over x_4."},{"Start":"00:53.140 ","End":"00:55.485","Text":"On the denominator,"},{"Start":"00:55.485 ","End":"00:57.635","Text":"x_5 is the highest power,"},{"Start":"00:57.635 ","End":"01:01.570","Text":"will take this outside the brackets, x_5,"},{"Start":"01:01.570 ","End":"01:08.520","Text":"and we\u0027re left with 3 plus 10 over x_4."},{"Start":"01:08.520 ","End":"01:14.700","Text":"Now, we can cancel x_4 cancels because x_5 is higher so what I\u0027ll just do,"},{"Start":"01:14.700 ","End":"01:18.140","Text":"the answer will be 1 over x. I\u0027ll just cross out this,"},{"Start":"01:18.140 ","End":"01:20.460","Text":"and now cross out the 5,"},{"Start":"01:20.460 ","End":"01:23.460","Text":"which means that we have x in the denominator."},{"Start":"01:23.460 ","End":"01:28.625","Text":"What we get now we can substitute x equals infinity."},{"Start":"01:28.625 ","End":"01:34.175","Text":"What we get is 1 plus 2 over"},{"Start":"01:34.175 ","End":"01:40.325","Text":"infinity squared plus 6 over infinity to the fourth,"},{"Start":"01:40.325 ","End":"01:49.520","Text":"all over x, which is the infinity times 3 plus 10 over infinity to the fourth."},{"Start":"01:49.520 ","End":"01:53.045","Text":"Now, infinity to any power is infinity,"},{"Start":"01:53.045 ","End":"01:57.995","Text":"so all I have to remind you is that if I have some number,"},{"Start":"01:57.995 ","End":"02:00.845","Text":"regular number a over infinity,"},{"Start":"02:00.845 ","End":"02:02.915","Text":"that could be plus or minus,"},{"Start":"02:02.915 ","End":"02:04.985","Text":"that\u0027s equal to 0."},{"Start":"02:04.985 ","End":"02:07.100","Text":"In this case, this is going to be 0,"},{"Start":"02:07.100 ","End":"02:08.525","Text":"this is going to be 0."},{"Start":"02:08.525 ","End":"02:19.430","Text":"We\u0027re left with 1 plus 0 plus 0 over infinity times 3 plus 0."},{"Start":"02:19.430 ","End":"02:25.055","Text":"1 plus 0 plus 0 equals 1 and 3 times infinity,"},{"Start":"02:25.055 ","End":"02:27.935","Text":"any positive number times infinity is still infinity,"},{"Start":"02:27.935 ","End":"02:31.490","Text":"1 over infinity, and this equals 0."},{"Start":"02:31.490 ","End":"02:34.170","Text":"That\u0027s our answer. We\u0027re done."}],"ID":4760},{"Watched":false,"Name":"Exercise 6","Duration":"4m 24s","ChapterTopicVideoID":4752,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4752.jpeg","UploadDate":"2016-05-02T11:23:07.1930000","DurationForVideoObject":"PT4M24S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.320","Text":"In this exercise,"},{"Start":"00:01.320 ","End":"00:06.135","Text":"we have to compute the limit as x goes to infinity of this whole expression."},{"Start":"00:06.135 ","End":"00:12.135","Text":"Now, if we try to do it naively by just substituting x equals infinity,"},{"Start":"00:12.135 ","End":"00:19.710","Text":"we\u0027ll get something of the form infinity over infinity minus infinity,"},{"Start":"00:19.710 ","End":"00:22.470","Text":"which is 1 of those undefined indeterminate."},{"Start":"00:22.470 ","End":"00:24.600","Text":"Just this part alone we can\u0027t"},{"Start":"00:24.600 ","End":"00:29.610","Text":"compute and also if it was infinity minus infinity, we also couldn\u0027t do it."},{"Start":"00:29.610 ","End":"00:33.330","Text":"So we\u0027re going to have to do some algebraic manipulation for us and"},{"Start":"00:33.330 ","End":"00:37.200","Text":"our old tricks of taking out factors. Let\u0027s see."},{"Start":"00:37.200 ","End":"00:41.630","Text":"The second thing to do would be to take a factor out,"},{"Start":"00:41.630 ","End":"00:46.190","Text":"but I can\u0027t even do that because there\u0027s a difference subtraction here."},{"Start":"00:46.190 ","End":"00:50.390","Text":"The best thing is to put this whole thing over a common denominator."},{"Start":"00:50.390 ","End":"00:51.755","Text":"Let\u0027s start writing."},{"Start":"00:51.755 ","End":"00:56.705","Text":"The limit as x goes to infinity of,"},{"Start":"00:56.705 ","End":"00:58.790","Text":"let\u0027s see what a common denominator could be."},{"Start":"00:58.790 ","End":"01:01.120","Text":"Well, 2 goes into 2x plus 10."},{"Start":"01:01.120 ","End":"01:07.665","Text":"So we can just make 2x plus 10, the common denominator."},{"Start":"01:07.665 ","End":"01:10.770","Text":"Here we\u0027re left with just as it is,"},{"Start":"01:10.770 ","End":"01:18.420","Text":"x squared minus 5x plus 6 minus."},{"Start":"01:18.420 ","End":"01:21.480","Text":"But here, we\u0027re going to, because this is 2,"},{"Start":"01:21.480 ","End":"01:25.460","Text":"2 goes into 2x plus 10 this thing goes into this."},{"Start":"01:25.460 ","End":"01:26.270","Text":"I\u0027ll just write it here."},{"Start":"01:26.270 ","End":"01:29.670","Text":"X plus 5 times."},{"Start":"01:29.810 ","End":"01:34.790","Text":"Usually indicate how many times this went into the common denominator."},{"Start":"01:34.790 ","End":"01:36.755","Text":"This whole thing times 1 is what it is,"},{"Start":"01:36.755 ","End":"01:39.095","Text":"and this thing times x, x plus 5."},{"Start":"01:39.095 ","End":"01:48.930","Text":"So minus x times x plus 5 and close the brackets."},{"Start":"01:48.930 ","End":"01:53.200","Text":"Let\u0027s see, let\u0027s do the algebra and we\u0027ll get,"},{"Start":"01:53.200 ","End":"01:56.675","Text":"just scroll down a bit to give us more space."},{"Start":"01:56.675 ","End":"01:58.385","Text":"This will equal,"},{"Start":"01:58.385 ","End":"02:04.390","Text":"limit x goes to infinity of,"},{"Start":"02:04.390 ","End":"02:07.580","Text":"now let\u0027s see if we can do some of this in our heads."},{"Start":"02:07.580 ","End":"02:11.240","Text":"This thing here, this part here I\u0027ll just write it above,"},{"Start":"02:11.240 ","End":"02:18.330","Text":"is minus x squared minus 5x."},{"Start":"02:18.740 ","End":"02:23.310","Text":"If I combine the minus x squared and the x squared"},{"Start":"02:23.310 ","End":"02:27.570","Text":"cancel and the minus 5x and minus 5x is reinforced."},{"Start":"02:27.570 ","End":"02:30.100","Text":"I get minus 10x,"},{"Start":"02:30.470 ","End":"02:34.540","Text":"and then the plus 6 from here."},{"Start":"02:35.030 ","End":"02:41.920","Text":"All over 2x plus 10."},{"Start":"02:42.170 ","End":"02:44.660","Text":"I could cancel by 2,"},{"Start":"02:44.660 ","End":"02:47.585","Text":"but that\u0027s not going to help us here."},{"Start":"02:47.585 ","End":"02:49.475","Text":"The thing to do now,"},{"Start":"02:49.475 ","End":"02:53.524","Text":"is to use our trick of taking outside the brackets,"},{"Start":"02:53.524 ","End":"02:55.760","Text":"the highest power of x,"},{"Start":"02:55.760 ","End":"02:58.510","Text":"both in the numerator and in the denominator."},{"Start":"02:58.510 ","End":"03:03.815","Text":"What we get, the highest power of x is just x but if we take it out,"},{"Start":"03:03.815 ","End":"03:10.710","Text":"we get x minus 10 plus 6 over x,"},{"Start":"03:10.710 ","End":"03:13.385","Text":"and on the denominator,"},{"Start":"03:13.385 ","End":"03:20.910","Text":"we\u0027ll get x times 2 plus 10 over x."},{"Start":"03:20.910 ","End":"03:24.540","Text":"The limit as x goes to infinity."},{"Start":"03:24.540 ","End":"03:28.920","Text":"This equals, lucky that the x cancels with"},{"Start":"03:28.920 ","End":"03:33.260","Text":"the x and what we\u0027re left with if we just substitute x equals"},{"Start":"03:33.260 ","End":"03:42.670","Text":"infinity is minus 10 plus 6 over infinity"},{"Start":"03:42.670 ","End":"03:51.765","Text":"over 2 plus 10 over infinity."},{"Start":"03:51.765 ","End":"03:54.720","Text":"Now I\u0027ve mentioned that several times."},{"Start":"03:54.720 ","End":"03:56.810","Text":"I can mention it again,"},{"Start":"03:56.810 ","End":"04:02.570","Text":"that whenever we have a number and it\u0027s divided by infinity,"},{"Start":"04:02.570 ","End":"04:04.755","Text":"but it could be minus infinity,"},{"Start":"04:04.755 ","End":"04:07.110","Text":"this thing is equal to 0."},{"Start":"04:07.110 ","End":"04:15.230","Text":"I can use that both here and here and what we\u0027ll get will be minus"},{"Start":"04:15.230 ","End":"04:24.870","Text":"10 plus 0 over 2 plus 0 and that comes out to be"}],"ID":4761},{"Watched":false,"Name":"Exercise 7","Duration":"2m 30s","ChapterTopicVideoID":4753,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4753.jpeg","UploadDate":"2016-05-01T15:17:39.7570000","DurationForVideoObject":"PT2M30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.280","Text":"In this exercise, we have to find the limit as x goes to infinity of this expression."},{"Start":"00:05.280 ","End":"00:07.380","Text":"It\u0027s little bit different from before."},{"Start":"00:07.380 ","End":"00:09.765","Text":"We\u0027re used to polynomial over polynomial,"},{"Start":"00:09.765 ","End":"00:12.480","Text":"as usually you try substituting x equals infinity"},{"Start":"00:12.480 ","End":"00:15.720","Text":"first and infinity squared plus 1 is infinity."},{"Start":"00:15.720 ","End":"00:19.845","Text":"Long story short, we get something of the form infinity over infinity,"},{"Start":"00:19.845 ","End":"00:21.300","Text":"which is not good to us."},{"Start":"00:21.300 ","End":"00:26.475","Text":"We\u0027re going to have to use some small tricks algebraic manipulation."},{"Start":"00:26.475 ","End":"00:30.150","Text":"What we\u0027re going to do is something similar to what we did with polynomials."},{"Start":"00:30.150 ","End":"00:32.020","Text":"We\u0027ll take a common factor out."},{"Start":"00:32.020 ","End":"00:34.595","Text":"We\u0027ll start with under the square root sign."},{"Start":"00:34.595 ","End":"00:36.720","Text":"Let\u0027s first of all, write it,"},{"Start":"00:36.720 ","End":"00:41.975","Text":"lim as x goes to infinity of the square root,"},{"Start":"00:41.975 ","End":"00:44.840","Text":"and I\u0027ll fill this in in a minute, over x."},{"Start":"00:44.840 ","End":"00:48.580","Text":"What I\u0027m going to do is take x squared outside the brackets."},{"Start":"00:48.580 ","End":"00:51.180","Text":"I\u0027ll take x squared here,"},{"Start":"00:51.180 ","End":"00:56.045","Text":"and what I\u0027m left with is 1 minus 1 over x squared."},{"Start":"00:56.045 ","End":"00:58.730","Text":"Now, if you remember your algebra,"},{"Start":"00:58.730 ","End":"01:03.185","Text":"and I will just write down this basic algebraic fact at the side."},{"Start":"01:03.185 ","End":"01:06.530","Text":"The square root of a times b,"},{"Start":"01:06.530 ","End":"01:09.124","Text":"assuming that a and b are both positive,"},{"Start":"01:09.124 ","End":"01:14.660","Text":"is equal to the square root of a times the square root of b."},{"Start":"01:14.660 ","End":"01:16.834","Text":"If I do that here,"},{"Start":"01:16.834 ","End":"01:22.775","Text":"what I\u0027ll get is the limit x goes to infinity."},{"Start":"01:22.775 ","End":"01:26.555","Text":"Now the square root of x squared is just x,"},{"Start":"01:26.555 ","End":"01:28.325","Text":"and the square root of the other part,"},{"Start":"01:28.325 ","End":"01:29.975","Text":"I\u0027ll just write it as it is,"},{"Start":"01:29.975 ","End":"01:36.170","Text":"all over x. I\u0027ll just make another comment that the square root of x squared,"},{"Start":"01:36.170 ","End":"01:39.320","Text":"it\u0027s not always x, it\u0027s usually the absolute value of x,"},{"Start":"01:39.320 ","End":"01:43.530","Text":"but it\u0027s equal to x if x is positive."},{"Start":"01:43.530 ","End":"01:45.080","Text":"Now it\u0027s going to infinity,"},{"Start":"01:45.080 ","End":"01:46.775","Text":"so it\u0027s certainly positive."},{"Start":"01:46.775 ","End":"01:50.080","Text":"So that\u0027s why it\u0027s x and not absolute value of x."},{"Start":"01:50.080 ","End":"01:52.335","Text":"Continuing, and look at that,"},{"Start":"01:52.335 ","End":"01:54.300","Text":"we can cancel the x."},{"Start":"01:54.300 ","End":"01:59.450","Text":"At this point, we can substitute x equals infinity and get"},{"Start":"01:59.450 ","End":"02:05.800","Text":"the square root of 1 minus 1 over infinity squared."},{"Start":"02:05.800 ","End":"02:11.705","Text":"Again, I want to remind you of something that when we have a over infinity,"},{"Start":"02:11.705 ","End":"02:14.915","Text":"where a is some regular number not infinity,"},{"Start":"02:14.915 ","End":"02:17.814","Text":"this is equal to 0."},{"Start":"02:17.814 ","End":"02:20.880","Text":"Here, we have 1 over infinity."},{"Start":"02:20.880 ","End":"02:22.380","Text":"So that\u0027s 0."},{"Start":"02:22.380 ","End":"02:26.135","Text":"So this is equal to 1 minus 0, the square root of 1."},{"Start":"02:26.135 ","End":"02:28.475","Text":"The answer is just 1,"},{"Start":"02:28.475 ","End":"02:30.720","Text":"and that\u0027s the answer."}],"ID":4762},{"Watched":false,"Name":"Exercise 8","Duration":"2m 49s","ChapterTopicVideoID":4754,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4754.jpeg","UploadDate":"2016-05-01T15:17:54.6970000","DurationForVideoObject":"PT2M49S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.615","Text":"This exercise is very similar to the previous exercise,"},{"Start":"00:03.615 ","End":"00:05.910","Text":"where we had x going to infinity,"},{"Start":"00:05.910 ","End":"00:09.645","Text":"on here we have x tending to minus infinity."},{"Start":"00:09.645 ","End":"00:12.885","Text":"As before, if we do a straightforward substitution,"},{"Start":"00:12.885 ","End":"00:19.820","Text":"we get something of the form infinity over infinity or minus infinity."},{"Start":"00:19.820 ","End":"00:21.680","Text":"So we have to use techniques."},{"Start":"00:21.680 ","End":"00:25.340","Text":"We\u0027ll use similar techniques as before by taking x"},{"Start":"00:25.340 ","End":"00:29.095","Text":"squared outside the brackets under the square root sign."},{"Start":"00:29.095 ","End":"00:38.660","Text":"We get the limit as x goes to minus infinity of the square root of x squared,"},{"Start":"00:38.660 ","End":"00:40.580","Text":"I\u0027m taking out the brackets here,"},{"Start":"00:40.580 ","End":"00:46.060","Text":"1 plus 1 over x squared over x."},{"Start":"00:46.060 ","End":"00:48.185","Text":"Once again, like before,"},{"Start":"00:48.185 ","End":"00:49.505","Text":"I\u0027m going to split it up."},{"Start":"00:49.505 ","End":"00:54.105","Text":"The square root of a product is square root of each 1 separately."},{"Start":"00:54.105 ","End":"01:02.525","Text":"This is equal to the limit as x goes to minus infinity of the square root of"},{"Start":"01:02.525 ","End":"01:11.750","Text":"x squared times the square root of 1 plus 1 over x squared all this over x."},{"Start":"01:11.750 ","End":"01:17.725","Text":"Whereas previously, I had said that this was just x and we can cancel x with x."},{"Start":"01:17.725 ","End":"01:21.634","Text":"This time, there\u0027s a slight catch here because"},{"Start":"01:21.634 ","End":"01:27.080","Text":"the square root of x squared generally is equal to the absolute value of x."},{"Start":"01:27.080 ","End":"01:28.550","Text":"If x is negative,"},{"Start":"01:28.550 ","End":"01:31.010","Text":"the absolute value of x is minus x."},{"Start":"01:31.010 ","End":"01:36.500","Text":"In our case, the square root of x squared over x,"},{"Start":"01:36.500 ","End":"01:39.755","Text":"it\u0027s actually minus x over x,"},{"Start":"01:39.755 ","End":"01:42.140","Text":"so it\u0027s equal to minus 1."},{"Start":"01:42.140 ","End":"01:44.660","Text":"So here it doesn\u0027t exactly cancel."},{"Start":"01:44.660 ","End":"01:47.150","Text":"What it does is it gives us a minus sign."},{"Start":"01:47.150 ","End":"01:52.730","Text":"We have the limit as x goes to minus infinity"},{"Start":"01:52.730 ","End":"01:58.655","Text":"of minus the square root of 1 plus 1 over x squared."},{"Start":"01:58.655 ","End":"02:02.210","Text":"Then I also have to remind you that if a is a number then"},{"Start":"02:02.210 ","End":"02:08.155","Text":"a over infinity could be plus or minus is equal to 0."},{"Start":"02:08.155 ","End":"02:13.955","Text":"Now, we can substitute x equals minus infinity is just minus"},{"Start":"02:13.955 ","End":"02:21.105","Text":"the square root of 1 plus 1 over minus infinity squared,"},{"Start":"02:21.105 ","End":"02:24.335","Text":"and minus infinity squared is plus infinity."},{"Start":"02:24.335 ","End":"02:26.090","Text":"Either way plus or minus infinity,"},{"Start":"02:26.090 ","End":"02:28.449","Text":"1 over this thing is 0."},{"Start":"02:28.449 ","End":"02:33.675","Text":"We get minus the square root of 1 plus 0,"},{"Start":"02:33.675 ","End":"02:37.060","Text":"which is just minus 1,"},{"Start":"02:37.060 ","End":"02:39.095","Text":"and that\u0027s the answer."},{"Start":"02:39.095 ","End":"02:43.940","Text":"Just watch out for that square root of x squared when x is negative."},{"Start":"02:43.940 ","End":"02:46.460","Text":"Of course, I should have mentioned that it\u0027s negative because"},{"Start":"02:46.460 ","End":"02:50.040","Text":"it\u0027s going to minus infinity. We\u0027re done."}],"ID":4763},{"Watched":false,"Name":"Exercise 9","Duration":"2m 55s","ChapterTopicVideoID":4755,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4755.jpeg","UploadDate":"2016-05-01T15:18:10.2930000","DurationForVideoObject":"PT2M55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.430","Text":"In this exercise, we have to find the limit as"},{"Start":"00:02.430 ","End":"00:05.025","Text":"x goes to minus infinity of this expression,"},{"Start":"00:05.025 ","End":"00:07.170","Text":"which has a square root in the numerator."},{"Start":"00:07.170 ","End":"00:11.610","Text":"As usual, we first try substituting minus infinity, but if you do that,"},{"Start":"00:11.610 ","End":"00:13.380","Text":"you\u0027re just going to get an expression of"},{"Start":"00:13.380 ","End":"00:17.655","Text":"the form infinity over infinity or plus or minus,"},{"Start":"00:17.655 ","End":"00:21.065","Text":"probably minus, but it\u0027s indeterminate,"},{"Start":"00:21.065 ","End":"00:24.710","Text":"undefined, and so we\u0027re going to have to use some matrix."},{"Start":"00:24.710 ","End":"00:26.090","Text":"Matrix we have used before,"},{"Start":"00:26.090 ","End":"00:29.560","Text":"which is taking out a power of x under the square root."},{"Start":"00:29.560 ","End":"00:36.215","Text":"What we\u0027ll get is the limit as x goes to minus infinity, square root,"},{"Start":"00:36.215 ","End":"00:41.385","Text":"and we\u0027ll take x^6 outside the brackets and that will leave us"},{"Start":"00:41.385 ","End":"00:47.610","Text":"with 9 minus 5x over x^6 is x^5."},{"Start":"00:47.610 ","End":"00:50.930","Text":"That\u0027s the numerator and on the denominator,"},{"Start":"00:50.930 ","End":"00:52.705","Text":"we take the x cubed out."},{"Start":"00:52.705 ","End":"00:54.799","Text":"We get x cubed,"},{"Start":"00:54.799 ","End":"01:01.100","Text":"1 minus 2 over x plus 1 over x cubed."},{"Start":"01:01.100 ","End":"01:02.960","Text":"Here something to watch out for,"},{"Start":"01:02.960 ","End":"01:04.070","Text":"we are going to of course,"},{"Start":"01:04.070 ","End":"01:11.495","Text":"use the formula where the square root of ab equals square root of a,"},{"Start":"01:11.495 ","End":"01:14.210","Text":"square root of b and in this case,"},{"Start":"01:14.210 ","End":"01:17.120","Text":"we\u0027re going to have to deal with the square root of x^6,"},{"Start":"01:17.120 ","End":"01:21.170","Text":"and normally the square root of something squared,"},{"Start":"01:21.170 ","End":"01:26.660","Text":"this is the square root of x cubed squared."},{"Start":"01:26.660 ","End":"01:28.880","Text":"You would think it would be x cubed,"},{"Start":"01:28.880 ","End":"01:32.420","Text":"but you should remember that it\u0027s absolute value of x cubed."},{"Start":"01:32.420 ","End":"01:34.880","Text":"Now, x is going to minus infinity,"},{"Start":"01:34.880 ","End":"01:36.785","Text":"so it certainly negative."},{"Start":"01:36.785 ","End":"01:40.850","Text":"So x cubed is also negative because negative cubed is negative."},{"Start":"01:40.850 ","End":"01:44.690","Text":"What this means that the absolute value of a negative quantity"},{"Start":"01:44.690 ","End":"01:48.890","Text":"is its negation is this is equal to minus x cubed."},{"Start":"01:48.890 ","End":"01:51.050","Text":"With this in mind,"},{"Start":"01:51.050 ","End":"01:55.340","Text":"limit as x goes to minus infinity,"},{"Start":"01:55.340 ","End":"02:01.355","Text":"we get the square root of x^6 over x cubed."},{"Start":"02:01.355 ","End":"02:10.760","Text":"That continuing, we also get the square root of 9 minus 5 over x^5 times this thing."},{"Start":"02:10.760 ","End":"02:15.770","Text":"Now, this over this doesn\u0027t exactly cancel because what we saw there,"},{"Start":"02:15.770 ","End":"02:17.975","Text":"but we do get a minus."},{"Start":"02:17.975 ","End":"02:21.770","Text":"The other thing I wanted to remind you of is that if we have a number a"},{"Start":"02:21.770 ","End":"02:25.670","Text":"over whether it\u0027s plus infinity or minus infinity,"},{"Start":"02:25.670 ","End":"02:27.230","Text":"this is equal to 0."},{"Start":"02:27.230 ","End":"02:29.630","Text":"We\u0027re going to use that here, here and here."},{"Start":"02:29.630 ","End":"02:34.790","Text":"What we end up getting is this is equal to minus, from here,"},{"Start":"02:34.790 ","End":"02:43.580","Text":"the square root of 9 minus 0 over 1 minus 0 plus 0."},{"Start":"02:43.580 ","End":"02:46.160","Text":"In short, the square root of 9 is 3,"},{"Start":"02:46.160 ","End":"02:52.760","Text":"the denominator\u0027s 1 minus 3 over 1 is just minus 3."},{"Start":"02:52.760 ","End":"02:55.440","Text":"That\u0027s all there is to it."}],"ID":4764},{"Watched":false,"Name":"Exercise 10","Duration":"4m 47s","ChapterTopicVideoID":4756,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4756.jpeg","UploadDate":"2016-05-30T05:11:45.3500000","DurationForVideoObject":"PT4M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.735","Text":"In this exercise, we have to find the limit as x goes to infinity"},{"Start":"00:03.735 ","End":"00:08.760","Text":"of the cube root of some polynomial over the square root of another polynomial."},{"Start":"00:08.760 ","End":"00:11.535","Text":"We\u0027re going to use our usual bag of tricks,"},{"Start":"00:11.535 ","End":"00:16.410","Text":"which is taking out under the root sign the highest power of x."},{"Start":"00:16.410 ","End":"00:25.350","Text":"Let\u0027s just rewrite this as the limit as x goes to infinity,"},{"Start":"00:25.350 ","End":"00:30.300","Text":"and the numerator, the cube root of,"},{"Start":"00:30.300 ","End":"00:32.895","Text":"now the highest power of x, not the first one,"},{"Start":"00:32.895 ","End":"00:37.920","Text":"is x^6, so let\u0027s write down x^6,"},{"Start":"00:37.920 ","End":"00:41.285","Text":"and this, we have to take outside the brackets."},{"Start":"00:41.285 ","End":"00:50.555","Text":"What we\u0027re left with is x^4 over x^6 is 1 over x squared."},{"Start":"00:50.555 ","End":"00:59.120","Text":"Here taking out x^6 gives us 2 over x^4."},{"Start":"00:59.120 ","End":"01:07.480","Text":"Here 6 over x^6, and the last one,"},{"Start":"01:07.480 ","End":"01:10.440","Text":"just 27 on its own,"},{"Start":"01:10.440 ","End":"01:12.815","Text":"and now the denominator,"},{"Start":"01:12.815 ","End":"01:17.930","Text":"the square root, the highest power here is x^4,"},{"Start":"01:17.930 ","End":"01:21.470","Text":"so it\u0027s x^4,"},{"Start":"01:21.470 ","End":"01:23.985","Text":"and when we take it out,"},{"Start":"01:23.985 ","End":"01:29.805","Text":"we get 3x cubed over x^4 is 3 over x."},{"Start":"01:29.805 ","End":"01:34.575","Text":"10 is going to be over x cubed,"},{"Start":"01:34.575 ","End":"01:38.080","Text":"and here, just 4 on its own."},{"Start":"01:38.080 ","End":"01:42.355","Text":"Now I want to remind you again that we are going to use some formulas,"},{"Start":"01:42.355 ","End":"01:52.105","Text":"1 is that the square root of ab is the square root of a square root of b,"},{"Start":"01:52.105 ","End":"01:55.210","Text":"and that\u0027s provided that a and b are positive,"},{"Start":"01:55.210 ","End":"02:01.030","Text":"and if they\u0027re positive, the same thing holds for the cube root of ab."},{"Start":"02:01.030 ","End":"02:04.900","Text":"Let\u0027s just write that down here."},{"Start":"02:04.900 ","End":"02:08.980","Text":"There\u0027s a restriction that a and b are positive for the cube root of a b is always"},{"Start":"02:08.980 ","End":"02:13.360","Text":"equal to the cube root of a cube root of b,"},{"Start":"02:13.360 ","End":"02:16.090","Text":"whether a and b are positive or not."},{"Start":"02:16.090 ","End":"02:19.915","Text":"So that\u0027s 1 thing. I know that we\u0027re also going to use the fact that"},{"Start":"02:19.915 ","End":"02:26.005","Text":"a number over plus or minus infinity is 0."},{"Start":"02:26.005 ","End":"02:34.975","Text":"What we get is the limit x goes to infinity,"},{"Start":"02:34.975 ","End":"02:38.120","Text":"we get the cube root"},{"Start":"02:38.120 ","End":"02:48.720","Text":"of x^6 times the cube root of all the rest of it,"},{"Start":"02:48.720 ","End":"03:00.165","Text":"which is 1 over x squared plus 2 over x^4 plus 6 over x^6 plus 27."},{"Start":"03:00.165 ","End":"03:10.900","Text":"On the denominator, the square root of x^4 times the square root of,"},{"Start":"03:10.900 ","End":"03:14.605","Text":"here I forgot to close the bracket."},{"Start":"03:14.605 ","End":"03:19.720","Text":"Here we get the 3 over x"},{"Start":"03:19.850 ","End":"03:27.075","Text":"plus 10 over x cubed plus 4."},{"Start":"03:27.075 ","End":"03:31.670","Text":"Now at the side here what I\u0027m going to do,"},{"Start":"03:31.670 ","End":"03:36.980","Text":"the cube root of x^6 is always equal"},{"Start":"03:36.980 ","End":"03:43.070","Text":"to x^6 over 3 it\u0027s x squared."},{"Start":"03:43.070 ","End":"03:49.010","Text":"The square root of x^4 is x squared squared,"},{"Start":"03:49.010 ","End":"03:54.725","Text":"is actually equal to the absolute value of x squared,"},{"Start":"03:54.725 ","End":"03:59.300","Text":"but since x is turning to infinity it\u0027s positive,"},{"Start":"03:59.300 ","End":"04:02.720","Text":"so it\u0027s actually equal to x squared."},{"Start":"04:02.720 ","End":"04:06.680","Text":"What we can do here is just cancel these 2 things."},{"Start":"04:06.680 ","End":"04:09.905","Text":"This is x squared and this is x squared."},{"Start":"04:09.905 ","End":"04:15.379","Text":"The next thing we\u0027ll do is use this fact that a number over infinity is 0."},{"Start":"04:15.379 ","End":"04:18.110","Text":"That will give us 0 here, here,"},{"Start":"04:18.110 ","End":"04:21.805","Text":"and here, and here, and here."},{"Start":"04:21.805 ","End":"04:25.415","Text":"What we\u0027re left with if we substitute x is infinity,"},{"Start":"04:25.415 ","End":"04:30.225","Text":"is just the cube root of"},{"Start":"04:30.225 ","End":"04:37.340","Text":"27 over the square root of 4."},{"Start":"04:37.340 ","End":"04:40.565","Text":"Cube root of 27 is 3,"},{"Start":"04:40.565 ","End":"04:42.560","Text":"square root of 4 is 2,"},{"Start":"04:42.560 ","End":"04:47.670","Text":"the answer is 3 over 2, and we\u0027re done."}],"ID":4765},{"Watched":false,"Name":"Exercise 11","Duration":"4m 20s","ChapterTopicVideoID":4757,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4757.jpeg","UploadDate":"2016-05-17T05:42:25.6870000","DurationForVideoObject":"PT4M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.300","Text":"In this exercise, we have to find the limit as x goes to"},{"Start":"00:03.300 ","End":"00:06.885","Text":"infinity of this expression involving square roots."},{"Start":"00:06.885 ","End":"00:10.830","Text":"Notice that all the expressions under the square root are linear."},{"Start":"00:10.830 ","End":"00:13.650","Text":"They all have the highest power is x."},{"Start":"00:13.650 ","End":"00:16.920","Text":"The usual technique of putting x equals infinity won\u0027t work."},{"Start":"00:16.920 ","End":"00:20.190","Text":"Infinity minus infinity is undefined and so on."},{"Start":"00:20.190 ","End":"00:24.900","Text":"We\u0027re going to have to use our familiar algebraic tricks,"},{"Start":"00:24.900 ","End":"00:32.040","Text":"and we\u0027re going to write this as is the limit of course x goes to infinity."},{"Start":"00:32.040 ","End":"00:35.555","Text":"In each case, the highest power of x is just x itself."},{"Start":"00:35.555 ","End":"00:44.045","Text":"We\u0027ll write it out as the square root of x times 1 plus 2 over x"},{"Start":"00:44.045 ","End":"00:54.785","Text":"minus the square root of x times 3 minus 3 over x."},{"Start":"00:54.785 ","End":"00:58.115","Text":"That\u0027s the numerator, the denominator we have."},{"Start":"00:58.115 ","End":"01:00.320","Text":"Again, we take x outside the brackets,"},{"Start":"01:00.320 ","End":"01:05.040","Text":"we\u0027re left with 4 plus 1 over x."},{"Start":"01:05.040 ","End":"01:15.905","Text":"Here we get the square root of x times 5 minus 1 over x."},{"Start":"01:15.905 ","End":"01:21.160","Text":"What I\u0027m going to do, we\u0027re going to use our usual formula, for example,"},{"Start":"01:21.160 ","End":"01:27.175","Text":"that the square root of ab is equal to the square root of a,"},{"Start":"01:27.175 ","End":"01:28.570","Text":"square root of b,"},{"Start":"01:28.570 ","End":"01:31.000","Text":"provided a and b are positive."},{"Start":"01:31.000 ","End":"01:40.605","Text":"The other formula that we\u0027re going to use is that a number a divided by infinity is 0."},{"Start":"01:40.605 ","End":"01:49.105","Text":"What we get is the limit x goes to infinity of the square root of x,"},{"Start":"01:49.105 ","End":"01:58.440","Text":"square root of 1 plus 2 over x minus the square root of x,"},{"Start":"01:58.440 ","End":"01:59.920","Text":"oh, here forgotten, the bracket,"},{"Start":"01:59.920 ","End":"02:05.780","Text":"times the square root of 3 minus 3 over x."},{"Start":"02:05.780 ","End":"02:10.189","Text":"All this over square root of x,"},{"Start":"02:10.189 ","End":"02:15.440","Text":"square root of 4 plus 1 over x minus finally,"},{"Start":"02:15.440 ","End":"02:21.500","Text":"square root of x, square root of 5 minus 1 over x."},{"Start":"02:21.500 ","End":"02:23.285","Text":"Now if you notice,"},{"Start":"02:23.285 ","End":"02:26.420","Text":"we have the square root of x in a lot of places."},{"Start":"02:26.420 ","End":"02:30.470","Text":"Now, if I took the square root of x outside the brackets,"},{"Start":"02:30.470 ","End":"02:33.440","Text":"I get a difference here of these 2 and the difference"},{"Start":"02:33.440 ","End":"02:36.935","Text":"here of these 2 and I could actually cancel top and bottom."},{"Start":"02:36.935 ","End":"02:43.070","Text":"But those of you who are a little bit rusty with your algebra, I\u0027ll just say that,"},{"Start":"02:43.070 ","End":"02:53.290","Text":"this looks something like ab minus ac over ad minus ae."},{"Start":"02:53.290 ","End":"02:56.600","Text":"What we would do, we take a outside the brackets"},{"Start":"02:56.600 ","End":"02:58.310","Text":"and get b minus c."},{"Start":"02:58.310 ","End":"03:00.050","Text":"Here we take a outside"},{"Start":"03:00.050 ","End":"03:03.980","Text":"the brackets and get d minus e."},{"Start":"03:03.980 ","End":"03:06.965","Text":"You can easily see that we would cancel the a."},{"Start":"03:06.965 ","End":"03:10.910","Text":"We might as well have canceled it in the first place over here."},{"Start":"03:10.910 ","End":"03:13.910","Text":"This equals limit x goes to infinity."},{"Start":"03:13.910 ","End":"03:15.770","Text":"Like I said, I\u0027m canceling this,"},{"Start":"03:15.770 ","End":"03:17.450","Text":"which is not the normal cancellation,"},{"Start":"03:17.450 ","End":"03:19.805","Text":"but here\u0027s my justification."},{"Start":"03:19.805 ","End":"03:25.400","Text":"Just write it as square root of 1 plus 2"},{"Start":"03:25.400 ","End":"03:31.600","Text":"over x minus the square root of 3 minus 3 over x."},{"Start":"03:31.600 ","End":"03:34.560","Text":"The stuff keeps repeating,"},{"Start":"03:34.560 ","End":"03:39.255","Text":"the set 5 minus 1 over x."},{"Start":"03:39.255 ","End":"03:43.645","Text":"Now if I put x is as infinity,"},{"Start":"03:43.645 ","End":"03:47.690","Text":"this is the formula that a over infinity is 0."},{"Start":"03:47.690 ","End":"03:49.670","Text":"I\u0027m going to get a 0 here."},{"Start":"03:49.670 ","End":"03:54.440","Text":"I\u0027m going to get a 0 here and here, and here."},{"Start":"03:54.440 ","End":"04:02.360","Text":"What I\u0027m going to be left with is from here I\u0027ll get a square root of 1."},{"Start":"04:02.360 ","End":"04:06.350","Text":"From here I get square root of 3."},{"Start":"04:06.350 ","End":"04:09.245","Text":"I\u0027ll copy the minus."},{"Start":"04:09.245 ","End":"04:13.380","Text":"From here I get the square root of 5."},{"Start":"04:13.420 ","End":"04:17.180","Text":"From here I get the square root of 4."},{"Start":"04:17.180 ","End":"04:20.549","Text":"This basically is our answer."}],"ID":4766},{"Watched":false,"Name":"Exercise 12","Duration":"6m 32s","ChapterTopicVideoID":4758,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4758.jpeg","UploadDate":"2016-05-17T05:43:05.7700000","DurationForVideoObject":"PT6M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.050","Text":"In this exercise, we have to find the limit as"},{"Start":"00:04.050 ","End":"00:08.055","Text":"x goes to minus infinity of this whole expression."},{"Start":"00:08.055 ","End":"00:12.900","Text":"This is very similar to the previous exercise only there we had infinity."},{"Start":"00:12.900 ","End":"00:16.020","Text":"The first thing to do with infinity is to try and substitute."},{"Start":"00:16.020 ","End":"00:20.025","Text":"I\u0027d like to just remind you of a formula."},{"Start":"00:20.025 ","End":"00:27.855","Text":"The first one is that a^minus infinity is equal to 0,"},{"Start":"00:27.855 ","End":"00:32.445","Text":"provided that a is bigger than 1."},{"Start":"00:32.445 ","End":"00:36.980","Text":"If we try substituting x is minus infinity,"},{"Start":"00:36.980 ","End":"00:40.640","Text":"these exponents are all minus infinity also."},{"Start":"00:40.640 ","End":"00:45.665","Text":"According to this, it will be 0 so we get 0 plus 0 plus 0 plus 0."},{"Start":"00:45.665 ","End":"00:50.810","Text":"In short, we get one of those undefined indeterminate forms,"},{"Start":"00:50.810 ","End":"00:52.290","Text":"0 over 0,"},{"Start":"00:52.290 ","End":"00:58.295","Text":"so we\u0027ll have to use some algebraic techniques to simplify this."},{"Start":"00:58.295 ","End":"00:59.970","Text":"This is a problem with exponents,"},{"Start":"00:59.970 ","End":"01:03.560","Text":"so you might as well remember some of the rules for the exponents."},{"Start":"01:03.560 ","End":"01:08.160","Text":"The main ones are that a^b plus"},{"Start":"01:08.160 ","End":"01:14.650","Text":"c is actually the product a^b times a^c."},{"Start":"01:14.650 ","End":"01:18.500","Text":"The other formula that\u0027s very similar is with a minus,"},{"Start":"01:18.500 ","End":"01:23.975","Text":"in which case we get a^b over a^c."},{"Start":"01:23.975 ","End":"01:28.530","Text":"The third useful formula is a power of a power,"},{"Start":"01:29.260 ","End":"01:35.360","Text":"a^b^c is just a^bc."},{"Start":"01:35.360 ","End":"01:37.760","Text":"We have different bases here."},{"Start":"01:37.760 ","End":"01:38.960","Text":"This is a base 2,"},{"Start":"01:38.960 ","End":"01:41.420","Text":"base 2, base 4, base 16,"},{"Start":"01:41.420 ","End":"01:49.880","Text":"but we could convert them all to base 2 because 16 is 2^4,"},{"Start":"01:49.880 ","End":"01:53.970","Text":"and 4 is 2^2."},{"Start":"01:54.320 ","End":"01:57.765","Text":"We could rewrite this limit,"},{"Start":"01:57.765 ","End":"02:00.470","Text":"x goes to"},{"Start":"02:00.470 ","End":"02:08.930","Text":"minus infinity of 2^4^x."},{"Start":"02:08.930 ","End":"02:10.010","Text":"I use this formula,"},{"Start":"02:10.010 ","End":"02:12.565","Text":"so it\u0027s 4 times x."},{"Start":"02:12.565 ","End":"02:18.630","Text":"The second one here is 2^2^x plus 1,"},{"Start":"02:18.630 ","End":"02:21.015","Text":"so it\u0027s 2 times x plus 1."},{"Start":"02:21.015 ","End":"02:22.470","Text":"Twice x plus 1,"},{"Start":"02:22.470 ","End":"02:25.545","Text":"I\u0027ll write it straight away as 2x plus 2."},{"Start":"02:25.545 ","End":"02:31.200","Text":"Denominator, 2^4x plus"},{"Start":"02:31.200 ","End":"02:38.090","Text":"2 as is and 2^x plus 3 as is."},{"Start":"02:38.090 ","End":"02:40.700","Text":"Now in case you did the previous exercise,"},{"Start":"02:40.700 ","End":"02:44.540","Text":"you\u0027ll note that there we took 2^4x outside the brackets,"},{"Start":"02:44.540 ","End":"02:46.190","Text":"and that seemed to work for us,"},{"Start":"02:46.190 ","End":"02:47.735","Text":"here it won\u0027t work."},{"Start":"02:47.735 ","End":"02:51.230","Text":"Trial and error shows that you should actually take this outside"},{"Start":"02:51.230 ","End":"02:55.355","Text":"the brackets here and this also in the denominator."},{"Start":"02:55.355 ","End":"02:58.160","Text":"In this case, what\u0027s going to work is taking this outside"},{"Start":"02:58.160 ","End":"03:02.090","Text":"the brackets in the numerator and here in the denominator."},{"Start":"03:02.090 ","End":"03:08.925","Text":"Continuing, we get the limit as x goes to minus infinity."},{"Start":"03:08.925 ","End":"03:15.000","Text":"Like I said, here we take 2x plus 2 outside the brackets"},{"Start":"03:15.000 ","End":"03:21.900","Text":"and here I\u0027ll take 2^x plus 3 outside the brackets."},{"Start":"03:21.900 ","End":"03:24.660","Text":"We have to do here,"},{"Start":"03:24.660 ","End":"03:29.055","Text":"2_4x over 2^2x plus 2."},{"Start":"03:29.055 ","End":"03:36.800","Text":"We need to use this formula and do here we have 2^4x"},{"Start":"03:36.800 ","End":"03:41.660","Text":"over 2^2x plus 2"},{"Start":"03:41.660 ","End":"03:48.739","Text":"is 2^4x minus 2x plus 2,"},{"Start":"03:48.739 ","End":"03:57.840","Text":"which is 2^4x minus 2x is 2x and minus the 2."},{"Start":"03:57.840 ","End":"04:04.980","Text":"So here we have 2^2x minus 2."},{"Start":"04:04.980 ","End":"04:07.570","Text":"The next one is just 1."},{"Start":"04:07.570 ","End":"04:09.110","Text":"Here we have a similar thing."},{"Start":"04:09.110 ","End":"04:13.970","Text":"We have to do, 2^4x plus"},{"Start":"04:13.970 ","End":"04:19.490","Text":"2 over 2^x plus 3,"},{"Start":"04:19.490 ","End":"04:28.850","Text":"which is 2^4x plus 2 less x plus 3,"},{"Start":"04:28.850 ","End":"04:34.530","Text":"which is 2^4x minus x is 3x,"},{"Start":"04:34.530 ","End":"04:39.730","Text":"2 minus 3 is minus 1."},{"Start":"04:39.730 ","End":"04:47.190","Text":"Here we get 2^3x minus 1."},{"Start":"04:47.190 ","End":"04:49.950","Text":"Here we just get 1."},{"Start":"04:49.950 ","End":"04:52.100","Text":"Now we do a little bit of canceling,"},{"Start":"04:52.100 ","End":"04:53.210","Text":"this over this,"},{"Start":"04:53.210 ","End":"04:57.300","Text":"we use the formula that this over this equals this."},{"Start":"04:57.300 ","End":"05:01.740","Text":"We have to do 2x plus 2 less x plus 3,"},{"Start":"05:01.740 ","End":"05:11.295","Text":"and that will give us 2x minus x is x and 2 minus 3 is minus 1."},{"Start":"05:11.295 ","End":"05:14.910","Text":"I\u0027ll put that outside the fraction."},{"Start":"05:14.910 ","End":"05:22.485","Text":"Here we get 2^2x minus 2 plus"},{"Start":"05:22.485 ","End":"05:32.135","Text":"1 over 2^3x minus 1 plus 1."},{"Start":"05:32.135 ","End":"05:36.545","Text":"Now notice that when x goes to minus infinity,"},{"Start":"05:36.545 ","End":"05:41.405","Text":"that all these expressions also go to minus infinity."},{"Start":"05:41.405 ","End":"05:46.585","Text":"Minus infinity takeaway 1 is minus infinity twice minus infinity,"},{"Start":"05:46.585 ","End":"05:49.290","Text":"because 2 is positive is also minus infinity,"},{"Start":"05:49.290 ","End":"05:53.615","Text":"and the numbers minus 2 doesn\u0027t change it and this is minus infinity."},{"Start":"05:53.615 ","End":"05:58.670","Text":"Basically all these things are minus infinity and we\u0027re going to use"},{"Start":"05:58.670 ","End":"06:04.805","Text":"this formula because 2 is bigger than 1 so the power of minus infinity is 0."},{"Start":"06:04.805 ","End":"06:07.220","Text":"It\u0027s equal when we put minus infinity in,"},{"Start":"06:07.220 ","End":"06:10.475","Text":"2^minus infinity is 0,"},{"Start":"06:10.475 ","End":"06:12.240","Text":"this would be 0,"},{"Start":"06:12.240 ","End":"06:15.105","Text":"this is plus 1,"},{"Start":"06:15.105 ","End":"06:18.545","Text":"this is also 0 plus 1."},{"Start":"06:18.545 ","End":"06:20.760","Text":"Anyway, 0 times something,"},{"Start":"06:20.760 ","End":"06:24.725","Text":"as long as this is not 0 in the denominator,"},{"Start":"06:24.725 ","End":"06:32.160","Text":"the whole thing comes out to be just equal to 0 and that\u0027s the answer."}],"ID":4767},{"Watched":false,"Name":"Exercise 13","Duration":"4m 59s","ChapterTopicVideoID":4759,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4759.jpeg","UploadDate":"2016-05-17T05:43:37.2500000","DurationForVideoObject":"PT4M59S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.565","Text":"In this exercise, we have to find the limit of x goes to infinity of this expression,"},{"Start":"00:05.565 ","End":"00:09.120","Text":"which involves the exponents and the fraction."},{"Start":"00:09.120 ","End":"00:12.015","Text":"We\u0027ve seen this exercise before,"},{"Start":"00:12.015 ","End":"00:20.115","Text":"and the direct substitution of infinity just leads us to infinity over infinity,"},{"Start":"00:20.115 ","End":"00:23.070","Text":"so we can\u0027t just substitute,"},{"Start":"00:23.070 ","End":"00:26.895","Text":"we\u0027re going to have to do some algebra here and simplify."},{"Start":"00:26.895 ","End":"00:30.375","Text":"I\u0027m going to remind you of some of the basic rules of"},{"Start":"00:30.375 ","End":"00:34.125","Text":"exponents that you should be pretty familiar with them by now."},{"Start":"00:34.125 ","End":"00:44.230","Text":"That a^b plus c is a^b times a^c."},{"Start":"00:44.230 ","End":"00:47.255","Text":"Also, if we have a minus here,"},{"Start":"00:47.255 ","End":"00:53.710","Text":"it turns into a quotient, a^b over a^c."},{"Start":"00:53.710 ","End":"00:58.610","Text":"If we have an exponent of an exponent a^b^c,"},{"Start":"00:58.610 ","End":"01:04.140","Text":"that\u0027s a to the power of bc."},{"Start":"01:04.140 ","End":"01:06.920","Text":"If we notice the basis here are 3,"},{"Start":"01:06.920 ","End":"01:09.335","Text":"3, 9 and 81,"},{"Start":"01:09.335 ","End":"01:14.360","Text":"all of these can be expressed in terms of base 3 because the"},{"Start":"01:14.360 ","End":"01:22.875","Text":"9 is just 3 squared and the 81 is 3 to the fourth."},{"Start":"01:22.875 ","End":"01:27.230","Text":"So if I use these and substitute in here,"},{"Start":"01:27.230 ","End":"01:33.095","Text":"what I will get is the limit x goes to infinity."},{"Start":"01:33.095 ","End":"01:35.220","Text":"The 4 is just there."},{"Start":"01:35.220 ","End":"01:39.620","Text":"9^x is 3^2^x."},{"Start":"01:39.620 ","End":"01:44.670","Text":"Using this formula, next 3, I\u0027ll leave alone."},{"Start":"01:44.670 ","End":"01:46.995","Text":"X plus 1,"},{"Start":"01:46.995 ","End":"01:49.980","Text":"81 is 3 to the fourth,"},{"Start":"01:49.980 ","End":"01:53.235","Text":"and it\u0027s to the power of 0.5x,"},{"Start":"01:53.235 ","End":"01:58.770","Text":"so I get 3^4"},{"Start":"01:58.770 ","End":"02:05.190","Text":"times 0.5x."},{"Start":"02:05.190 ","End":"02:07.380","Text":"Here 3 is what we want,"},{"Start":"02:07.380 ","End":"02:12.574","Text":"so it just leaves it, 3^x plus 3."},{"Start":"02:12.574 ","End":"02:16.055","Text":"The next step is to take something outside the brackets,"},{"Start":"02:16.055 ","End":"02:19.580","Text":"and we behave differently with infinity as with minus infinity."},{"Start":"02:19.580 ","End":"02:21.125","Text":"Where it\u0027s plus infinity,"},{"Start":"02:21.125 ","End":"02:28.515","Text":"we generally take the larger of the coefficients on the 2x and the x plus 1,"},{"Start":"02:28.515 ","End":"02:30.030","Text":"the 2 is larger,"},{"Start":"02:30.030 ","End":"02:33.430","Text":"so that\u0027s what we take out is 3^2x."},{"Start":"02:33.430 ","End":"02:38.580","Text":"Here I\u0027ll just point out to the side here that 4 times"},{"Start":"02:38.580 ","End":"02:44.090","Text":"0.5 is 2 instead of that, write 2x."},{"Start":"02:44.090 ","End":"02:46.790","Text":"Here we have 2x and x plus 3, again,"},{"Start":"02:46.790 ","End":"02:51.035","Text":"we\u0027ll choose the 2x to take outside the brackets."},{"Start":"02:51.035 ","End":"02:59.480","Text":"What we get is the limit as x goes to infinity."},{"Start":"02:59.480 ","End":"03:03.860","Text":"Now here we take 3^2x outside,"},{"Start":"03:03.860 ","End":"03:09.770","Text":"and in the denominator also 3^2x we take outside the brackets."},{"Start":"03:09.770 ","End":"03:11.135","Text":"Let\u0027s see what we\u0027re left with."},{"Start":"03:11.135 ","End":"03:12.760","Text":"Here we have a 4,"},{"Start":"03:12.760 ","End":"03:16.580","Text":"here we use this rule here of the subtraction,"},{"Start":"03:16.580 ","End":"03:22.610","Text":"we need x plus 1 minus 2x is 1 minus x."},{"Start":"03:22.610 ","End":"03:30.695","Text":"So that\u0027s 3^1 minus x."},{"Start":"03:30.695 ","End":"03:34.820","Text":"Now the denominator, we\u0027ve taken out 3^2x,"},{"Start":"03:34.820 ","End":"03:37.315","Text":"so here we\u0027re just left with 1,"},{"Start":"03:37.315 ","End":"03:40.905","Text":"and if we take out 2x from here,"},{"Start":"03:40.905 ","End":"03:51.130","Text":"so it\u0027s 3 instead of 1 minus x here we have 3 minus x, and this cancels."},{"Start":"03:52.040 ","End":"03:58.435","Text":"This point, all we have to do is substitute x equals infinity,"},{"Start":"03:58.435 ","End":"04:00.670","Text":"and what we get is,"},{"Start":"04:00.670 ","End":"04:09.190","Text":"from here, this 4 is this 4, 3^1 minus infinity."},{"Start":"04:09.190 ","End":"04:13.180","Text":"From the denominator, we get a 1, just copy it,"},{"Start":"04:13.180 ","End":"04:19.150","Text":"and 3^3 minus infinity."},{"Start":"04:19.150 ","End":"04:23.750","Text":"Another thing that I should have mentioned is that a to"},{"Start":"04:23.750 ","End":"04:29.330","Text":"the power of minus infinity is equal to 0,"},{"Start":"04:29.330 ","End":"04:31.610","Text":"provided that a is bigger than 1."},{"Start":"04:31.610 ","End":"04:33.815","Text":"For example, if a is 3,"},{"Start":"04:33.815 ","End":"04:40.310","Text":"1 minus infinity is minus infinity because adding the 1 doesn\u0027t change an infinity."},{"Start":"04:40.310 ","End":"04:44.045","Text":"Same here, 3 to the minus infinity."},{"Start":"04:44.045 ","End":"04:53.730","Text":"What we get is just 4 plus 0 over 1 plus 0,"},{"Start":"04:53.730 ","End":"04:57.750","Text":"and that boils down to just 4."},{"Start":"04:57.750 ","End":"05:00.160","Text":"That\u0027s our answer."}],"ID":4768},{"Watched":false,"Name":"Exercise 14","Duration":"4m 44s","ChapterTopicVideoID":4760,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4760.jpeg","UploadDate":"2016-05-17T05:44:06.2630000","DurationForVideoObject":"PT4M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.565","Text":"This exercise is the same as the previous one,"},{"Start":"00:03.565 ","End":"00:08.340","Text":"except that we have a minus infinity instead of infinity."},{"Start":"00:08.340 ","End":"00:11.650","Text":"So we will reuse some of the techniques of the previous exercise,"},{"Start":"00:11.650 ","End":"00:14.015","Text":"but there will also be differences."},{"Start":"00:14.015 ","End":"00:16.250","Text":"The first thing I\u0027d like to mention,"},{"Start":"00:16.250 ","End":"00:19.570","Text":"because we\u0027re going to try and substitute x equals minus infinity,"},{"Start":"00:19.570 ","End":"00:25.665","Text":"is that a to the power of minus infinity is 0, so I emphasize."},{"Start":"00:25.665 ","End":"00:27.400","Text":"The other thing is that we\u0027re going to use"},{"Start":"00:27.400 ","End":"00:30.595","Text":"the same trick as before of using the common base 3."},{"Start":"00:30.595 ","End":"00:38.860","Text":"We can do this because 9 is 3 to the power of 2 and 81 is 3 to the power of 4."},{"Start":"00:38.860 ","End":"00:43.000","Text":"Let\u0027s rewrite this limit in terms of base 3."},{"Start":"00:43.000 ","End":"00:49.335","Text":"Limit x goes to minus infinity."},{"Start":"00:49.335 ","End":"00:54.270","Text":"4 time 9 is 3 squared so the 2 goes"},{"Start":"00:54.270 ","End":"00:59.970","Text":"with the x so we get 3 to the power of 2x."},{"Start":"00:59.970 ","End":"01:09.450","Text":"Here, just the same 3 to the x plus 1 divided by 81 is 3 to the 4,"},{"Start":"01:09.450 ","End":"01:12.910","Text":"so we need to multiply 4 times this thing."},{"Start":"01:12.910 ","End":"01:19.845","Text":"Well, 4 times a 1/2 is 2 so we get 3 to the 2x here,"},{"Start":"01:19.845 ","End":"01:21.885","Text":"and this stays the same,"},{"Start":"01:21.885 ","End":"01:25.615","Text":"3 to the x plus 3."},{"Start":"01:25.615 ","End":"01:27.770","Text":"Now, the difference between this and"},{"Start":"01:27.770 ","End":"01:30.755","Text":"the previous exercise is what we take outside the bracket."},{"Start":"01:30.755 ","End":"01:32.465","Text":"In the case of infinity,"},{"Start":"01:32.465 ","End":"01:35.225","Text":"it\u0027s the 2x that\u0027s chosen,"},{"Start":"01:35.225 ","End":"01:37.670","Text":"basically the one with the larger coefficient."},{"Start":"01:37.670 ","End":"01:39.815","Text":"But in the case of minus infinity,"},{"Start":"01:39.815 ","End":"01:42.035","Text":"we\u0027re going to choose the smaller one."},{"Start":"01:42.035 ","End":"01:46.640","Text":"We\u0027re going to choose here the x plus 1 and here the x plus 3."},{"Start":"01:46.640 ","End":"01:55.040","Text":"That may make this more precise limit x goes to minus infinity."},{"Start":"01:55.040 ","End":"02:02.050","Text":"Now at the top, we take out 3 to the x plus 1,"},{"Start":"02:02.050 ","End":"02:03.584","Text":"and at the bottom,"},{"Start":"02:03.584 ","End":"02:08.910","Text":"3 to the x plus 3,"},{"Start":"02:08.910 ","End":"02:11.540","Text":"and then inside the brackets,"},{"Start":"02:11.540 ","End":"02:13.370","Text":"the easy part is the last one,"},{"Start":"02:13.370 ","End":"02:16.170","Text":"here it\u0027s going to be plus 1 and"},{"Start":"02:16.170 ","End":"02:19.745","Text":"here it\u0027s going to be plus 1 because of what we took out."},{"Start":"02:19.745 ","End":"02:25.100","Text":"What we\u0027re left with is 3 to the 2x."},{"Start":"02:25.100 ","End":"02:28.730","Text":"We need to divide it by what we took out,"},{"Start":"02:28.730 ","End":"02:32.435","Text":"which is 3 to the x plus 1,"},{"Start":"02:32.435 ","End":"02:35.420","Text":"and that\u0027s 3 to the power of,"},{"Start":"02:35.420 ","End":"02:40.985","Text":"subtract 2x minus x plus 1,"},{"Start":"02:40.985 ","End":"02:47.120","Text":"which is 3 to the x minus 1."},{"Start":"02:47.120 ","End":"02:49.400","Text":"Let\u0027s not forget the 4."},{"Start":"02:49.400 ","End":"02:55.485","Text":"So 4 times 3 to the x minus 1."},{"Start":"02:55.485 ","End":"02:57.875","Text":"Down below something very similar,"},{"Start":"02:57.875 ","End":"03:06.245","Text":"we just have to do 2x minus x plus 3 and then we\u0027ll get 3 to the x minus 3."},{"Start":"03:06.245 ","End":"03:11.794","Text":"We can also cancel the 3 to the x plus 1 over 3 to the x plus 3."},{"Start":"03:11.794 ","End":"03:14.900","Text":"As before, we just need to subtract the exponents."},{"Start":"03:14.900 ","End":"03:23.205","Text":"That\u0027s this x plus 1 minus x plus 3 is just minus 2,"},{"Start":"03:23.205 ","End":"03:29.450","Text":"so what we get here if we do this division is"},{"Start":"03:29.450 ","End":"03:38.435","Text":"the limit x goes to minus infinity of 3 to the minus 2 is 1/9."},{"Start":"03:38.435 ","End":"03:46.455","Text":"So we get 1/9 and then 4 times 3 to the x"},{"Start":"03:46.455 ","End":"03:55.780","Text":"minus 1 plus 1 over 3 to the x minus 3 plus 1."},{"Start":"03:55.780 ","End":"04:01.990","Text":"At this point, we can just substitute x equals minus infinity, and remember,"},{"Start":"04:01.990 ","End":"04:07.250","Text":"minus infinity minus 1 is still minus infinity and if I subtract 3 from minus infinity,"},{"Start":"04:07.250 ","End":"04:11.840","Text":"it\u0027s still minus infinity and I\u0027m using this thing with a equals 3,"},{"Start":"04:11.840 ","End":"04:15.410","Text":"so this will be 0 and this will be 0."},{"Start":"04:15.410 ","End":"04:21.090","Text":"So what we get is 1/9,"},{"Start":"04:21.090 ","End":"04:30.035","Text":"4 times 0 plus 1 over 0 plus 1."},{"Start":"04:30.035 ","End":"04:33.065","Text":"Well, 4 time 0 plus 1 is just 1,"},{"Start":"04:33.065 ","End":"04:36.605","Text":"and this is also 1 times 1/9,"},{"Start":"04:36.605 ","End":"04:44.110","Text":"so the answer is just 1/9. Well done."}],"ID":4769},{"Watched":false,"Name":"Exercise 15","Duration":"2m 36s","ChapterTopicVideoID":4761,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4761.jpeg","UploadDate":"2016-05-17T05:44:22.2370000","DurationForVideoObject":"PT2M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.840","Text":"In this exercise, we have to find the limit as x goes to"},{"Start":"00:03.840 ","End":"00:09.210","Text":"infinity of the square root of polynomial over a polynomial."},{"Start":"00:09.210 ","End":"00:11.700","Text":"If we just try to substitute infinity,"},{"Start":"00:11.700 ","End":"00:15.960","Text":"we get infinity over infinity here."},{"Start":"00:15.960 ","End":"00:22.335","Text":"Basically, something like square root of infinity over infinity,"},{"Start":"00:22.335 ","End":"00:25.320","Text":"which is undefined, indeterminate."},{"Start":"00:25.320 ","End":"00:27.390","Text":"We have to use another trick,"},{"Start":"00:27.390 ","End":"00:33.960","Text":"and the idea here is that we\u0027re allowed to put the limit inside the square root."},{"Start":"00:33.960 ","End":"00:38.999","Text":"What I mean is that the limit as x goes to infinity of this thing is exactly"},{"Start":"00:38.999 ","End":"00:44.460","Text":"equal to the square root of the limit"},{"Start":"00:44.460 ","End":"00:51.570","Text":"as x goes to infinity of 4x squared"},{"Start":"00:51.570 ","End":"01:00.460","Text":"plus 2 over x squared plus 1000x."},{"Start":"01:02.330 ","End":"01:09.350","Text":"What I\u0027m going to do is just compute the inside without the square root,"},{"Start":"01:09.350 ","End":"01:14.255","Text":"just this part here as a side exercise,"},{"Start":"01:14.255 ","End":"01:17.300","Text":"and then we\u0027ll put it back up here."},{"Start":"01:17.300 ","End":"01:22.580","Text":"What we have is this limit here of the inside,"},{"Start":"01:22.580 ","End":"01:28.385","Text":"that\u0027s the limit as x goes to infinity."},{"Start":"01:28.385 ","End":"01:31.085","Text":"But this kind of exercise we\u0027ve done before,"},{"Start":"01:31.085 ","End":"01:35.360","Text":"we take out the highest power at the top and in the bottom."},{"Start":"01:35.360 ","End":"01:40.430","Text":"It\u0027s x squared in both cases so we take out the x squared."},{"Start":"01:40.430 ","End":"01:48.720","Text":"Here, we\u0027re left with 4 plus 2 over x squared, and here,"},{"Start":"01:48.720 ","End":"01:58.635","Text":"we\u0027re left with x squared times 1 plus 1000 over x."},{"Start":"01:58.635 ","End":"02:01.440","Text":"This thing cancels,"},{"Start":"02:01.440 ","End":"02:05.600","Text":"and we substitute x equals infinity here."},{"Start":"02:05.600 ","End":"02:08.675","Text":"Now, 2 over infinity is 0,"},{"Start":"02:08.675 ","End":"02:11.450","Text":"1000 over infinity is 0,"},{"Start":"02:11.450 ","End":"02:17.120","Text":"so basically, what we get here is just 4."},{"Start":"02:17.120 ","End":"02:21.515","Text":"4 plus 0 over 1 plus 0 is 4."},{"Start":"02:21.515 ","End":"02:24.370","Text":"But now, this was just the inside."},{"Start":"02:24.370 ","End":"02:27.770","Text":"If I put the 4 back into here,"},{"Start":"02:27.770 ","End":"02:31.970","Text":"what I\u0027m going to be left with is the square root of 4,"},{"Start":"02:31.970 ","End":"02:33.725","Text":"which is 2,"},{"Start":"02:33.725 ","End":"02:36.240","Text":"and that\u0027s our answer."}],"ID":4770},{"Watched":false,"Name":"Exercise 16","Duration":"3m 12s","ChapterTopicVideoID":4762,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4762.jpeg","UploadDate":"2016-05-17T05:44:42.0870000","DurationForVideoObject":"PT3M12S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.420","Text":"In this exercise, we have to find the limit of"},{"Start":"00:03.420 ","End":"00:08.400","Text":"the natural log of this expression, polynomial over polynomial."},{"Start":"00:08.400 ","End":"00:10.890","Text":"Now, if we didn\u0027t have the natural log,"},{"Start":"00:10.890 ","End":"00:12.990","Text":"we\u0027d know how to do this."},{"Start":"00:12.990 ","End":"00:17.700","Text":"The purpose of this exercise is to show that we can actually compute"},{"Start":"00:17.700 ","End":"00:22.870","Text":"the limit of what\u0027s inside the natural log and then apply the natural log."},{"Start":"00:22.870 ","End":"00:24.720","Text":"The first thing to try, of course,"},{"Start":"00:24.720 ","End":"00:27.255","Text":"would be to substitute x equals infinity."},{"Start":"00:27.255 ","End":"00:30.480","Text":"But then we would just get infinity over infinity,"},{"Start":"00:30.480 ","End":"00:34.225","Text":"which is one of those undefined forms."},{"Start":"00:34.225 ","End":"00:42.140","Text":"What we\u0027re going to do is say that this equals the natural logarithm of"},{"Start":"00:42.140 ","End":"00:50.590","Text":"the limit as x goes to infinity of just what\u0027s inside here."},{"Start":"00:50.590 ","End":"00:57.404","Text":"3x cubed minus 5x minus 1"},{"Start":"00:57.404 ","End":"01:05.845","Text":"over x cubed minus 2x squared plus 1."},{"Start":"01:05.845 ","End":"01:13.385","Text":"Now, if I take this and just to aside exercise of computing this part,"},{"Start":"01:13.385 ","End":"01:18.470","Text":"and then I\u0027ll put it back in here and we\u0027ll take the natural log."},{"Start":"01:18.470 ","End":"01:20.690","Text":"This part here,"},{"Start":"01:20.690 ","End":"01:23.220","Text":"we use the usual techniques,"},{"Start":"01:23.220 ","End":"01:26.010","Text":"let\u0027s call it Asterisk."},{"Start":"01:26.010 ","End":"01:30.375","Text":"The asterisk will be the limit."},{"Start":"01:30.375 ","End":"01:34.760","Text":"The usual way to do it is to take out the highest power."},{"Start":"01:34.760 ","End":"01:43.849","Text":"We take out the x cubed and we\u0027re left with 3 minus 5"},{"Start":"01:43.849 ","End":"01:53.010","Text":"over x squared minus 1 over x cubed over,"},{"Start":"01:53.010 ","End":"01:56.300","Text":"and here again, the highest power is x cubed."},{"Start":"01:56.300 ","End":"02:01.820","Text":"It\u0027s x cubed and what we\u0027re left with is 1 minus"},{"Start":"02:01.820 ","End":"02:07.820","Text":"2 over x plus 1 over x cubed."},{"Start":"02:07.820 ","End":"02:10.100","Text":"This cancels."},{"Start":"02:10.100 ","End":"02:19.040","Text":"Since any positive quantity a over infinity is equal to 0."},{"Start":"02:19.040 ","End":"02:22.250","Text":"This is true when a is positive."},{"Start":"02:22.250 ","End":"02:25.100","Text":"If we substitute x equals infinity,"},{"Start":"02:25.100 ","End":"02:27.080","Text":"this is going to be infinity,"},{"Start":"02:27.080 ","End":"02:30.710","Text":"and this is going to be infinity because squared or cubed is still infinity."},{"Start":"02:30.710 ","End":"02:35.860","Text":"In other words, this is going to be 0 and this is 0 and you can see also this and this."},{"Start":"02:35.860 ","End":"02:38.105","Text":"If we substitute now,"},{"Start":"02:38.105 ","End":"02:47.535","Text":"what we get is 3 minus 0 minus 0 over"},{"Start":"02:47.535 ","End":"02:57.435","Text":"1 minus 0 plus 0 and that is just equal to 3."},{"Start":"02:57.435 ","End":"03:00.735","Text":"Now that 3 is the asterisk."},{"Start":"03:00.735 ","End":"03:06.440","Text":"We put that back inside here and we get"},{"Start":"03:06.440 ","End":"03:12.840","Text":"that the answer is the natural log of 3. We\u0027re done."}],"ID":4771},{"Watched":false,"Name":"Exercise 17","Duration":"2m 58s","ChapterTopicVideoID":4763,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4763.jpeg","UploadDate":"2016-05-17T05:45:00.5800000","DurationForVideoObject":"PT2M58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.020 ","End":"00:03.990","Text":"Here, we have to find the limit as x goes to"},{"Start":"00:03.990 ","End":"00:08.025","Text":"infinity of e to the power of this whole thing."},{"Start":"00:08.025 ","End":"00:11.370","Text":"This exponent is something that we know how to deal with."},{"Start":"00:11.370 ","End":"00:14.205","Text":"It\u0027s a polynomial over a polynomial."},{"Start":"00:14.205 ","End":"00:18.870","Text":"The purpose of this exercise is to show that we"},{"Start":"00:18.870 ","End":"00:24.320","Text":"can leave the e outside and just figure out the limit of the exponent."},{"Start":"00:24.320 ","End":"00:28.370","Text":"I should first mention that if we just put x equals infinity here,"},{"Start":"00:28.370 ","End":"00:30.710","Text":"we\u0027d get infinity over infinity,"},{"Start":"00:30.710 ","End":"00:33.680","Text":"which is an undefined indeterminate form,"},{"Start":"00:33.680 ","End":"00:36.605","Text":"which is why we need to use this trick."},{"Start":"00:36.605 ","End":"00:47.330","Text":"What we get is that this limit is equal to e to the power of the limit as"},{"Start":"00:47.330 ","End":"00:54.180","Text":"x goes to infinity of x to the 4th plus 2x"},{"Start":"00:54.180 ","End":"01:02.605","Text":"squared plus 6 over 3x to the 4th plus 10x."},{"Start":"01:02.605 ","End":"01:08.855","Text":"In other words, if you take the limit of e to the power of something,"},{"Start":"01:08.855 ","End":"01:13.370","Text":"it\u0027s the same as e to the power of the limit of that same thing."},{"Start":"01:13.370 ","End":"01:16.195","Text":"Now, this part here,"},{"Start":"01:16.195 ","End":"01:18.300","Text":"we\u0027ll do as a side exercise."},{"Start":"01:18.300 ","End":"01:22.085","Text":"Let\u0027s call that an asterisk. Let\u0027s copy it."},{"Start":"01:22.085 ","End":"01:25.850","Text":"That usual technique is to take out the highest power of x."},{"Start":"01:25.850 ","End":"01:29.410","Text":"In this case, it\u0027s x to the 4th,"},{"Start":"01:29.410 ","End":"01:32.645","Text":"and here it\u0027s also going to be x to the 4th."},{"Start":"01:32.645 ","End":"01:38.045","Text":"What we\u0027re left with is 1 plus 2 over x squared,"},{"Start":"01:38.045 ","End":"01:41.455","Text":"6 over x to the 4th,"},{"Start":"01:41.455 ","End":"01:43.875","Text":"and on the denominator,"},{"Start":"01:43.875 ","End":"01:50.805","Text":"3 plus 10 over x cubed."},{"Start":"01:50.805 ","End":"01:55.680","Text":"This cancels with this and at this point,"},{"Start":"01:55.680 ","End":"01:58.860","Text":"we can substitute infinity."},{"Start":"01:58.860 ","End":"02:08.800","Text":"Remember that any number a positive over infinity is equal to 0."},{"Start":"02:08.800 ","End":"02:11.905","Text":"That\u0027s provided that a is positive,"},{"Start":"02:11.905 ","End":"02:15.594","Text":"and so we\u0027re going to get here 0,"},{"Start":"02:15.594 ","End":"02:20.245","Text":"because infinity to the power of 2 or 4 or 3 is going to be infinity."},{"Start":"02:20.245 ","End":"02:23.155","Text":"This, this, and this will be 0."},{"Start":"02:23.155 ","End":"02:28.040","Text":"What we\u0027re left with is 1 plus"},{"Start":"02:28.040 ","End":"02:36.290","Text":"0 plus 0 over 3 plus 0."},{"Start":"02:36.290 ","End":"02:40.480","Text":"Now this is equal to 1/3."},{"Start":"02:40.480 ","End":"02:44.060","Text":"Now remember that\u0027s just the asterisk part."},{"Start":"02:44.060 ","End":"02:50.420","Text":"We have to put this 1/3 back over here and the answer will"},{"Start":"02:50.420 ","End":"02:57.510","Text":"be e to the power of 1/3 and we\u0027re done."}],"ID":4772},{"Watched":false,"Name":"Exercise 18","Duration":"3m 44s","ChapterTopicVideoID":4764,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4764.jpeg","UploadDate":"2016-06-06T05:34:53.1930000","DurationForVideoObject":"PT3M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.320","Text":"In this exercise, we have to figure out the limit as x goes to minus infinity of"},{"Start":"00:05.320 ","End":"00:11.275","Text":"the sign of this rational function polynomial over polynomial."},{"Start":"00:11.275 ","End":"00:16.580","Text":"We first try naively to substitute minus infinity."},{"Start":"00:16.580 ","End":"00:20.965","Text":"If I put minus infinity in the numerator here,"},{"Start":"00:20.965 ","End":"00:23.110","Text":"it behaves like the highest power,"},{"Start":"00:23.110 ","End":"00:24.370","Text":"so it\u0027s like x^4,"},{"Start":"00:24.370 ","End":"00:25.855","Text":"it goes to infinity."},{"Start":"00:25.855 ","End":"00:28.840","Text":"The denominator behaves like minus infinity."},{"Start":"00:28.840 ","End":"00:31.195","Text":"We have one of those infinity over infinity."},{"Start":"00:31.195 ","End":"00:33.460","Text":"Can\u0027t do it by substitution."},{"Start":"00:33.460 ","End":"00:36.710","Text":"We\u0027re going to have to use some tricks."},{"Start":"00:36.710 ","End":"00:40.374","Text":"The usual one is when you have a sign,"},{"Start":"00:40.374 ","End":"00:45.119","Text":"is to exchange the limit with the sign."},{"Start":"00:45.119 ","End":"00:47.735","Text":"What we going to do first is,"},{"Start":"00:47.735 ","End":"00:51.635","Text":"say that this is the sign of"},{"Start":"00:51.635 ","End":"00:58.010","Text":"the limit as x goes to minus infinity of same thing,"},{"Start":"00:58.010 ","End":"01:02.360","Text":"just copy it, x^ 4 plus 2x squared plus 6"},{"Start":"01:02.360 ","End":"01:08.305","Text":"over 3x^5 plus 10x."},{"Start":"01:08.305 ","End":"01:12.490","Text":"That would do the limit of this and then AN will do the sign of what we get."},{"Start":"01:12.490 ","End":"01:16.715","Text":"What I\u0027d like to do is just do this bit as side exercise,"},{"Start":"01:16.715 ","End":"01:18.365","Text":"let\u0027s say I\u0027ll do it over here,"},{"Start":"01:18.365 ","End":"01:22.550","Text":"call it and I don\u0027t know asterisk and do the asterisk over here."},{"Start":"01:22.550 ","End":"01:27.955","Text":"What I\u0027ve got is the limit as x goes to minus infinity."},{"Start":"01:27.955 ","End":"01:33.170","Text":"I\u0027m going to use a standard trick with polynomials over polynomials."},{"Start":"01:33.170 ","End":"01:37.670","Text":"We take out the highest power in each of the numerator and denominator separately."},{"Start":"01:37.670 ","End":"01:40.705","Text":"In here, x^4 is the highest power."},{"Start":"01:40.705 ","End":"01:42.990","Text":"I\u0027ll take it outside the brackets in a moment."},{"Start":"01:42.990 ","End":"01:45.620","Text":"In here, it\u0027ll be x^5."},{"Start":"01:45.620 ","End":"01:47.075","Text":"Let\u0027s see what we\u0027re left with."},{"Start":"01:47.075 ","End":"01:51.750","Text":"Here we have 1 plus dividing by x^4,"},{"Start":"01:51.750 ","End":"01:54.345","Text":"that becomes 2 over x squared,"},{"Start":"01:54.345 ","End":"01:57.735","Text":"and here, 6 over x^4."},{"Start":"01:57.735 ","End":"02:00.120","Text":"Here, if I take x^5 out,"},{"Start":"02:00.120 ","End":"02:02.025","Text":"I\u0027m left with 3 here."},{"Start":"02:02.025 ","End":"02:04.905","Text":"X over x^5 is x^4,"},{"Start":"02:04.905 ","End":"02:07.890","Text":"so it\u0027s 10 over x^4."},{"Start":"02:07.890 ","End":"02:12.470","Text":"Now I can do the limit as x goes to minus infinity."},{"Start":"02:12.470 ","End":"02:14.245","Text":"First of all, note,"},{"Start":"02:14.245 ","End":"02:17.820","Text":"that I can cancel x^4 over x^5"},{"Start":"02:17.820 ","End":"02:20.460","Text":"just gives me 1 over x."},{"Start":"02:20.460 ","End":"02:23.115","Text":"I just have an x in the bottom."},{"Start":"02:23.115 ","End":"02:27.270","Text":"What I get is,"},{"Start":"02:27.270 ","End":"02:28.470","Text":"I can now do the limit."},{"Start":"02:28.470 ","End":"02:35.910","Text":"The 1 over x is like 1 over infinity minus infinity."},{"Start":"02:37.520 ","End":"02:40.470","Text":"I don\u0027t need the 1."},{"Start":"02:40.470 ","End":"02:44.505","Text":"What I have here is 1 plus,"},{"Start":"02:44.505 ","End":"02:47.780","Text":"now 2 over infinity squared or"},{"Start":"02:47.780 ","End":"02:51.590","Text":"minus infinity squared doesn\u0027t even matter, is going to be 0."},{"Start":"02:51.590 ","End":"02:54.320","Text":"A number over infinity is 0."},{"Start":"02:54.320 ","End":"02:58.310","Text":"Here also, we are going to get 0, 6 times 0."},{"Start":"02:58.310 ","End":"03:05.010","Text":"Here we\u0027re going to get 3 plus 0. Now here\u0027s no problem."},{"Start":"03:05.010 ","End":"03:06.650","Text":"This is 1, this is 3,"},{"Start":"03:06.650 ","End":"03:07.985","Text":"but here we\u0027re having,"},{"Start":"03:07.985 ","End":"03:09.350","Text":"I\u0027ll leave the 1 in again."},{"Start":"03:09.350 ","End":"03:15.095","Text":"Anyway, 1 over minus infinity is minus 1 over infinity is 0."},{"Start":"03:15.095 ","End":"03:20.180","Text":"I\u0027ll just emphasize that this is the important thing and this is 0"},{"Start":"03:20.180 ","End":"03:26.645","Text":"so what we get is 0 times 1 over 3, It\u0027s just 0."},{"Start":"03:26.645 ","End":"03:30.580","Text":"Now I\u0027m going to go back here and put the 0 in,"},{"Start":"03:30.580 ","End":"03:33.410","Text":"and so this equals to sine,"},{"Start":"03:33.410 ","End":"03:36.365","Text":"and this whole thing is replaced by 0,"},{"Start":"03:36.365 ","End":"03:39.335","Text":"and sine of 0 is 0."},{"Start":"03:39.335 ","End":"03:43.950","Text":"This is the answer to the question. We\u0027re done."}],"ID":4773},{"Watched":false,"Name":"Exercise 19","Duration":"4m 37s","ChapterTopicVideoID":4765,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4765.jpeg","UploadDate":"2016-05-17T05:50:18.1270000","DurationForVideoObject":"PT4M37S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.315","Text":"Here we have to find the limit as x goes to infinity of this expression."},{"Start":"00:06.315 ","End":"00:10.620","Text":"If you notice, it\u0027s exactly the same as the previous exercise"},{"Start":"00:10.620 ","End":"00:16.200","Text":"where k was equal to 5 and here it\u0027s just more general."},{"Start":"00:16.200 ","End":"00:20.600","Text":"I\u0027m going to go through this a bit quicker because we\u0027ve done it before."},{"Start":"00:20.600 ","End":"00:23.730","Text":"Trying to substitute x equals infinity doesn\u0027t work,"},{"Start":"00:23.730 ","End":"00:26.585","Text":"it just gives us infinity minus infinity,"},{"Start":"00:26.585 ","End":"00:29.360","Text":"which is undefined, could be anything."},{"Start":"00:29.360 ","End":"00:34.025","Text":"We\u0027re going to use some algebra and we\u0027re going to use conjugates."},{"Start":"00:34.025 ","End":"00:38.330","Text":"That I will remind you again that the conjugate,"},{"Start":"00:38.330 ","End":"00:42.964","Text":"if we have something of the form a minus b,"},{"Start":"00:42.964 ","End":"00:48.590","Text":"then its conjugate is a plus b and vice versa,"},{"Start":"00:48.590 ","End":"00:51.695","Text":"these 2 expressions are conjugates of each other,"},{"Start":"00:51.695 ","End":"00:55.020","Text":"and the useful thing is that if you multiply them,"},{"Start":"00:55.020 ","End":"00:58.080","Text":"a minus b times a plus b,"},{"Start":"00:58.080 ","End":"01:03.190","Text":"is a difference of squares formula and it comes out to be a squared minus b squared,"},{"Start":"01:03.190 ","End":"01:09.360","Text":"so that if a or b is the square root after squaring no longer is."},{"Start":"01:10.090 ","End":"01:16.565","Text":"The other thing that\u0027s going to come in handy for later on is that if you have a number"},{"Start":"01:16.565 ","End":"01:22.820","Text":"a and you divide it by infinity could be plus or minus infinity, that equals 0."},{"Start":"01:22.820 ","End":"01:24.920","Text":"That might also be useful."},{"Start":"01:24.920 ","End":"01:26.945","Text":"What we\u0027ll do is,"},{"Start":"01:26.945 ","End":"01:34.160","Text":"take the limit as x goes to infinity and multiply by the conjugate,"},{"Start":"01:34.160 ","End":"01:39.065","Text":"so we get the square root of"},{"Start":"01:39.065 ","End":"01:44.610","Text":"x squared plus kx minus x,"},{"Start":"01:44.610 ","End":"01:47.820","Text":"all this times the conjugate."},{"Start":"01:47.820 ","End":"01:55.429","Text":"Square root of x squared plus kx plus x,"},{"Start":"01:55.429 ","End":"01:57.770","Text":"just change the sign from minus to a plus."},{"Start":"01:57.770 ","End":"01:59.840","Text":"But I can\u0027t just multiply by something,"},{"Start":"01:59.840 ","End":"02:03.080","Text":"I changed the exercise, I\u0027ve to compensate."},{"Start":"02:03.080 ","End":"02:12.970","Text":"If I divide, I multiplied by the square root of x squared plus kx plus x."},{"Start":"02:12.970 ","End":"02:16.790","Text":"If I cancel these 2 or multiplying by this over this which is 1,"},{"Start":"02:16.790 ","End":"02:19.640","Text":"so now I haven\u0027t changed the exercise."},{"Start":"02:19.640 ","End":"02:26.360","Text":"If I do the multiplication I just get this thing squared,"},{"Start":"02:26.360 ","End":"02:29.390","Text":"which is x squared plus kx minus x squared,"},{"Start":"02:29.390 ","End":"02:33.050","Text":"I\u0027m left in the numerator with just kx,"},{"Start":"02:33.050 ","End":"02:35.420","Text":"and then in the denominator,"},{"Start":"02:35.420 ","End":"02:41.450","Text":"we have the square root of x squared times"},{"Start":"02:41.450 ","End":"02:48.860","Text":"1 plus k over x, plus x."},{"Start":"02:48.860 ","End":"02:53.030","Text":"Now this equals the square root is,"},{"Start":"02:53.030 ","End":"02:56.000","Text":"by the rules of square root of a product is"},{"Start":"02:56.000 ","End":"03:00.005","Text":"just the product of the square roots provided that they\u0027re all positive."},{"Start":"03:00.005 ","End":"03:02.225","Text":"This is positive, of course,"},{"Start":"03:02.225 ","End":"03:10.030","Text":"and 1 plus k over x is also positive because k over x is pretty small."},{"Start":"03:10.040 ","End":"03:13.460","Text":"Even if k is negative, actually is big enough,"},{"Start":"03:13.460 ","End":"03:16.625","Text":"this thing is positive, so that\u0027s okay."},{"Start":"03:16.625 ","End":"03:22.110","Text":"The square root of x squared is the absolute value of x,"},{"Start":"03:22.110 ","End":"03:24.475","Text":"but when x is positive,"},{"Start":"03:24.475 ","End":"03:28.459","Text":"then the absolute value of x is just x itself."},{"Start":"03:28.459 ","End":"03:33.319","Text":"We end up getting this thing is x times"},{"Start":"03:33.319 ","End":"03:39.585","Text":"the square root of 1 plus k over x,"},{"Start":"03:39.585 ","End":"03:42.170","Text":"and then the x comes out of the brackets."},{"Start":"03:42.170 ","End":"03:47.595","Text":"Basically what I can do is cancel x with x and with x,"},{"Start":"03:47.595 ","End":"03:49.700","Text":"because x comes out of the brackets and cancels."},{"Start":"03:49.700 ","End":"03:56.630","Text":"What we\u0027re left with is the limit as x goes to infinity of"},{"Start":"03:56.630 ","End":"04:01.460","Text":"k over the square root"},{"Start":"04:01.460 ","End":"04:07.620","Text":"of 1 plus k over x plus 1."},{"Start":"04:07.620 ","End":"04:12.920","Text":"Now, what I mentioned here about a over infinity being 0 at this point,"},{"Start":"04:12.920 ","End":"04:15.200","Text":"our substitute x equals infinity,"},{"Start":"04:15.200 ","End":"04:17.090","Text":"and this part here,"},{"Start":"04:17.090 ","End":"04:20.045","Text":"this just becomes what I\u0027ve circled."},{"Start":"04:20.045 ","End":"04:23.135","Text":"I get square root of 1 is 1 plus 1 is 2,"},{"Start":"04:23.135 ","End":"04:28.830","Text":"so the answer would just be K over 2."},{"Start":"04:28.830 ","End":"04:30.525","Text":"That\u0027s our answer."},{"Start":"04:30.525 ","End":"04:32.990","Text":"I just got to show you that infinity minus"},{"Start":"04:32.990 ","End":"04:36.870","Text":"infinity can be practically anything. We\u0027re done."}],"ID":4774},{"Watched":false,"Name":"Exercise 20","Duration":"3m 29s","ChapterTopicVideoID":4766,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4766.jpeg","UploadDate":"2016-05-17T05:50:39.9600000","DurationForVideoObject":"PT3M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.830","Text":"Here we have to find the limit as x goes to infinity of this expression."},{"Start":"00:04.830 ","End":"00:07.935","Text":"We\u0027ve seen this thing many times before."},{"Start":"00:07.935 ","End":"00:11.430","Text":"A straight substitution gives infinity minus infinity,"},{"Start":"00:11.430 ","End":"00:15.330","Text":"and so we multiply and divide by the conjugate."},{"Start":"00:15.330 ","End":"00:23.400","Text":"What we get is that this is equal to the limit as x goes to"},{"Start":"00:23.400 ","End":"00:28.500","Text":"infinity of the square root of"},{"Start":"00:28.500 ","End":"00:34.260","Text":"x squared plus x plus 1 minus x,"},{"Start":"00:34.260 ","End":"00:38.775","Text":"and I\u0027m going to multiply this by its conjugate,"},{"Start":"00:38.775 ","End":"00:44.420","Text":"square root of x squared plus x plus 1 conjugate,"},{"Start":"00:44.420 ","End":"00:48.350","Text":"meaning we put plus x instead of minus x here."},{"Start":"00:48.350 ","End":"00:52.180","Text":"But we can\u0027t just multiply by something without dividing it,"},{"Start":"00:52.180 ","End":"00:54.665","Text":"so we also divide by the same thing."},{"Start":"00:54.665 ","End":"01:03.395","Text":"Square root of x squared plus x plus 1 plus x."},{"Start":"01:03.395 ","End":"01:09.920","Text":"This equals limit of this thing squared"},{"Start":"01:09.920 ","End":"01:18.524","Text":"is x squared plus x plus 1 minus this thing squared."},{"Start":"01:18.524 ","End":"01:21.495","Text":"Just as a quick reminder,"},{"Start":"01:21.495 ","End":"01:26.610","Text":"A minus B times its conjugate A plus B"},{"Start":"01:26.610 ","End":"01:30.155","Text":"is equal to A squared minus B squared."},{"Start":"01:30.155 ","End":"01:32.644","Text":"Famous formula from algebra."},{"Start":"01:32.644 ","End":"01:35.440","Text":"That\u0027s what I\u0027ve done in the numerator."},{"Start":"01:35.440 ","End":"01:37.225","Text":"On the denominator,"},{"Start":"01:37.225 ","End":"01:40.865","Text":"I\u0027m going to take x squared outside the brackets."},{"Start":"01:40.865 ","End":"01:43.670","Text":"Since we\u0027ve done this so many times before,"},{"Start":"01:43.670 ","End":"01:45.230","Text":"I\u0027ll cut out a step."},{"Start":"01:45.230 ","End":"01:50.655","Text":"So square root of x squared times the square root of,"},{"Start":"01:50.655 ","End":"01:52.580","Text":"when I take x squared out,"},{"Start":"01:52.580 ","End":"02:01.170","Text":"I\u0027m left with 1 plus 1 over x plus 1 over x squared plus x."},{"Start":"02:01.170 ","End":"02:04.900","Text":"The square root of x squared is equal"},{"Start":"02:04.900 ","End":"02:07.225","Text":"to the absolute value of x in general,"},{"Start":"02:07.225 ","End":"02:11.780","Text":"but is just equal to x when x is positive."},{"Start":"02:11.780 ","End":"02:14.844","Text":"Which it is because it\u0027s going to plus infinity."},{"Start":"02:14.844 ","End":"02:21.850","Text":"This is x equals limit x tends to infinity."},{"Start":"02:21.850 ","End":"02:24.519","Text":"I\u0027ll take x out of the numerator."},{"Start":"02:24.519 ","End":"02:28.670","Text":"Also, notice that this and this cancels."},{"Start":"02:28.670 ","End":"02:36.635","Text":"If I take x, I just get 1 plus 1 over x divided by,"},{"Start":"02:36.635 ","End":"02:40.535","Text":"now this thing, square root of x squared is just x."},{"Start":"02:40.535 ","End":"02:43.945","Text":"What we\u0027re left is if we take it outside the brackets,"},{"Start":"02:43.945 ","End":"02:47.900","Text":"is this square root here plus 1."},{"Start":"02:47.900 ","End":"02:49.445","Text":"I\u0027ll just copy what\u0027s here."},{"Start":"02:49.445 ","End":"02:55.055","Text":"1 plus 1 over x plus 1 over x squared."},{"Start":"02:55.055 ","End":"02:58.780","Text":"This x will cancel with this x,"},{"Start":"02:58.780 ","End":"03:04.190","Text":"and if we use the formula that any number"},{"Start":"03:04.190 ","End":"03:09.050","Text":"over infinity is equal to 0,"},{"Start":"03:09.050 ","End":"03:13.045","Text":"and here we\u0027ll get a 0, we get a 0 here and here."},{"Start":"03:13.045 ","End":"03:17.915","Text":"What we\u0027re going to be left with is a numerator 1,"},{"Start":"03:17.915 ","End":"03:19.565","Text":"and in the denominator,"},{"Start":"03:19.565 ","End":"03:22.220","Text":"square root of 1 plus 1 is 2,"},{"Start":"03:22.220 ","End":"03:25.640","Text":"so 1 over 2,"},{"Start":"03:25.640 ","End":"03:29.110","Text":"1/2 as the answer. We\u0027re done."}],"ID":4775},{"Watched":false,"Name":"Exercise 21","Duration":"5m 41s","ChapterTopicVideoID":4767,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4767.jpeg","UploadDate":"2016-05-30T05:12:23.5830000","DurationForVideoObject":"PT5M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.725","Text":"This exercise is very similar to the previous one,"},{"Start":"00:04.725 ","End":"00:06.480","Text":"where we had a minus x here,"},{"Start":"00:06.480 ","End":"00:08.685","Text":"and we had a plus infinity here."},{"Start":"00:08.685 ","End":"00:12.525","Text":"Here we have to find the limit as x goes to minus infinity of this."},{"Start":"00:12.525 ","End":"00:15.430","Text":"If we substitute minus infinity,"},{"Start":"00:15.430 ","End":"00:20.890","Text":"you\u0027re going to get into trouble because minus infinity squared is infinity."},{"Start":"00:20.890 ","End":"00:22.525","Text":"Here we have minus infinity."},{"Start":"00:22.525 ","End":"00:25.010","Text":"Already we have an infinity minus infinity"},{"Start":"00:25.010 ","End":"00:29.160","Text":"and no purpose in continuing to substitute anymore."},{"Start":"00:29.160 ","End":"00:33.170","Text":"We\u0027ll use the method of the conjugates as we did before."},{"Start":"00:33.170 ","End":"00:39.500","Text":"Just a quick reminder that the conjugate of A plus B,"},{"Start":"00:39.500 ","End":"00:45.180","Text":"this will be A, this will be B is A minus B, and vice versa."},{"Start":"00:45.180 ","End":"00:47.460","Text":"The conjugate of A minus B is a plus B."},{"Start":"00:47.460 ","End":"00:54.274","Text":"The advantage of using conjugates is that if you multiply an expression by its conjugate,"},{"Start":"00:54.274 ","End":"00:57.485","Text":"you get what is called the difference of squares formula."},{"Start":"00:57.485 ","End":"01:01.210","Text":"If either A or B is the square root after squaring it no longer is."},{"Start":"01:01.210 ","End":"01:05.060","Text":"The other thing that we\u0027re going to need is that any number"},{"Start":"01:05.060 ","End":"01:11.120","Text":"over infinity could be plus infinity or minus infinity is 0."},{"Start":"01:11.120 ","End":"01:20.000","Text":"Rewriting this thing, this equals the limit as x tends to"},{"Start":"01:20.000 ","End":"01:25.970","Text":"minus infinity of the square root of"},{"Start":"01:25.970 ","End":"01:32.970","Text":"x squared plus x plus 1 plus x."},{"Start":"01:32.970 ","End":"01:35.869","Text":"Since I can\u0027t just multiply by the conjugate,"},{"Start":"01:35.869 ","End":"01:37.220","Text":"I\u0027m going to have to compensate."},{"Start":"01:37.220 ","End":"01:39.800","Text":"But if I put the same thing on the top, and on the bottom,"},{"Start":"01:39.800 ","End":"01:45.395","Text":"I\u0027m multiplying by 1 times square root of"},{"Start":"01:45.395 ","End":"01:51.605","Text":"x squared plus x plus 1 this time minus x."},{"Start":"01:51.605 ","End":"01:53.930","Text":"The same thing on the denominator,"},{"Start":"01:53.930 ","End":"02:02.570","Text":"square root of x squared plus x plus 1 minus x,"},{"Start":"02:02.570 ","End":"02:09.785","Text":"which equals limit x goes to minus infinity."},{"Start":"02:09.785 ","End":"02:13.985","Text":"Multiplying this by this using the difference of squares formula,"},{"Start":"02:13.985 ","End":"02:22.320","Text":"we get x squared plus x plus 1 minus x squared."},{"Start":"02:22.320 ","End":"02:28.290","Text":"This thing cancels over the square root."},{"Start":"02:28.290 ","End":"02:31.150","Text":"We\u0027re going to use the same trick as we usually use."},{"Start":"02:31.150 ","End":"02:34.540","Text":"Take x squared outside the brackets under the square root and"},{"Start":"02:34.540 ","End":"02:40.165","Text":"then factor it as square root of x squared separately."},{"Start":"02:40.165 ","End":"02:41.770","Text":"The square root is"},{"Start":"02:41.770 ","End":"02:51.835","Text":"1 plus 1 over x plus 1 over x squared minus x."},{"Start":"02:51.835 ","End":"02:55.645","Text":"Then we take x outside the brackets,"},{"Start":"02:55.645 ","End":"03:02.270","Text":"and we\u0027re left with limit x goes to minus infinity."},{"Start":"03:02.270 ","End":"03:04.780","Text":"If I take x out of this,"},{"Start":"03:04.780 ","End":"03:11.620","Text":"I get 1 plus 1 over x divided by,"},{"Start":"03:11.620 ","End":"03:13.750","Text":"what is the square root of x squared?"},{"Start":"03:13.750 ","End":"03:20.350","Text":"The square root of x squared is the absolute value of x."},{"Start":"03:20.350 ","End":"03:23.485","Text":"But when x is negative,"},{"Start":"03:23.485 ","End":"03:26.860","Text":"even very negative, it\u0027s going to minus infinity."},{"Start":"03:26.860 ","End":"03:27.940","Text":"When x is negative,"},{"Start":"03:27.940 ","End":"03:31.450","Text":"the absolute value of x is minus x."},{"Start":"03:31.450 ","End":"03:35.485","Text":"For example, the square root of, say,"},{"Start":"03:35.485 ","End":"03:39.605","Text":"a million, which would be minus 1,000 squared,"},{"Start":"03:39.605 ","End":"03:41.640","Text":"would be plus 1,000."},{"Start":"03:41.640 ","End":"03:43.395","Text":"For x very negative,"},{"Start":"03:43.395 ","End":"03:45.810","Text":"we need to get something positive for the square roots."},{"Start":"03:45.810 ","End":"03:47.020","Text":"There\u0027s a minus here,"},{"Start":"03:47.020 ","End":"03:49.175","Text":"that\u0027s what you have to watch out for."},{"Start":"03:49.175 ","End":"03:55.510","Text":"Here we have minus x times same square root,"},{"Start":"03:55.510 ","End":"04:03.060","Text":"1 plus 1 over x plus 1 over x squared minus x."},{"Start":"04:03.060 ","End":"04:06.155","Text":"We\u0027ve done this thing before."},{"Start":"04:06.155 ","End":"04:09.830","Text":"I\u0027m going to take x out of the bottom also,"},{"Start":"04:09.830 ","End":"04:11.975","Text":"and then cancel the xs."},{"Start":"04:11.975 ","End":"04:18.800","Text":"I cancel the x from here together with the x from here and the x from here,"},{"Start":"04:18.800 ","End":"04:21.220","Text":"leaving it just with 1."},{"Start":"04:21.220 ","End":"04:27.875","Text":"Because I took x outside the brackets that get minus a square root minus 1."},{"Start":"04:27.875 ","End":"04:34.820","Text":"We get 1 plus 1 over x on the numerator."},{"Start":"04:34.820 ","End":"04:37.070","Text":"Now, the denominator,"},{"Start":"04:37.070 ","End":"04:43.055","Text":"square root of 1 plus 1 over x"},{"Start":"04:43.055 ","End":"04:48.260","Text":"plus 1 over x squared minus 1."},{"Start":"04:48.260 ","End":"04:51.980","Text":"Now I\u0027m going to use what I wrote here."},{"Start":"04:51.980 ","End":"04:56.045","Text":"Something over plus or minus infinity is 0."},{"Start":"04:56.045 ","End":"04:59.965","Text":"Here we have 1 over infinity and that\u0027s 0."},{"Start":"04:59.965 ","End":"05:03.560","Text":"At this point, we\u0027re going to actually substitute minus infinity."},{"Start":"05:03.560 ","End":"05:10.355","Text":"We get 1 plus 0 over"},{"Start":"05:10.355 ","End":"05:18.860","Text":"the square root of 1 plus 0 plus 0 minus 1."},{"Start":"05:18.860 ","End":"05:20.720","Text":"I forgot the minus here."},{"Start":"05:20.720 ","End":"05:22.580","Text":"This is minus here."},{"Start":"05:22.580 ","End":"05:28.535","Text":"Minus 1, could see something was going to be wrong if I did 1 minus 1 here."},{"Start":"05:28.535 ","End":"05:30.335","Text":"This is just 1."},{"Start":"05:30.335 ","End":"05:32.540","Text":"Here we have a square root of 1 is 1,"},{"Start":"05:32.540 ","End":"05:34.475","Text":"1 over minus 2,"},{"Start":"05:34.475 ","End":"05:38.340","Text":"in other words, minus a half."},{"Start":"05:38.340 ","End":"05:40.480","Text":"That\u0027s the answer."}],"ID":4776},{"Watched":false,"Name":"Exercise 22","Duration":"5m 33s","ChapterTopicVideoID":4768,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4768.jpeg","UploadDate":"2016-05-30T05:12:59.5430000","DurationForVideoObject":"PT5M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:02.340","Text":"In this exercise,"},{"Start":"00:02.340 ","End":"00:07.245","Text":"we have to find the limit as x goes to infinity of this expression."},{"Start":"00:07.245 ","End":"00:10.920","Text":"We\u0027ve done many of this kind before that if you just put infinity in here,"},{"Start":"00:10.920 ","End":"00:12.660","Text":"you get the square root of infinity,"},{"Start":"00:12.660 ","End":"00:17.100","Text":"which is infinity minus infinity and infinity minus"},{"Start":"00:17.100 ","End":"00:22.950","Text":"infinity is 1 of those undefined things and the answer could be anything."},{"Start":"00:22.950 ","End":"00:25.875","Text":"We use our usual tricks here."},{"Start":"00:25.875 ","End":"00:30.030","Text":"It\u0027s clear that we should use the conjugate because we have a square root here."},{"Start":"00:30.030 ","End":"00:33.450","Text":"I\u0027ll just give you a quick reminder of what a conjugate is."},{"Start":"00:33.450 ","End":"00:37.995","Text":"If we have an expression of the form A minus B,"},{"Start":"00:37.995 ","End":"00:44.340","Text":"then its conjugate is just change the sign or the other way around."},{"Start":"00:44.340 ","End":"00:48.330","Text":"These are conjugate to each other and the advantage of using this is that,"},{"Start":"00:48.330 ","End":"00:53.310","Text":"if we multiply A minus B by A plus B,"},{"Start":"00:53.310 ","End":"00:54.915","Text":"two conjugates together,"},{"Start":"00:54.915 ","End":"00:57.990","Text":"we get a difference of squares formula,"},{"Start":"00:57.990 ","End":"00:59.820","Text":"A squared minus B squared,"},{"Start":"00:59.820 ","End":"01:01.650","Text":"so if one of these is a square root,"},{"Start":"01:01.650 ","End":"01:05.340","Text":"when you square it, it will no longer be a square root."},{"Start":"01:05.340 ","End":"01:07.530","Text":"I\u0027ll just remind you that,"},{"Start":"01:07.530 ","End":"01:11.235","Text":"if I have a number A and I divide it by infinity,"},{"Start":"01:11.235 ","End":"01:13.230","Text":"could be plus or minus,"},{"Start":"01:13.230 ","End":"01:15.405","Text":"then that will be 0."},{"Start":"01:15.405 ","End":"01:23.370","Text":"So, the limit as x goes to infinity of"},{"Start":"01:23.370 ","End":"01:32.805","Text":"this square root of x to the fourth plus x squared plus 1 minus x squared."},{"Start":"01:32.805 ","End":"01:37.200","Text":"What I\u0027m going to do to make it over a fraction and multiply top and"},{"Start":"01:37.200 ","End":"01:41.760","Text":"bottom by the conjugate and if I multiply top and bottom by the same quantity,"},{"Start":"01:41.760 ","End":"01:43.530","Text":"it\u0027s like multiplying by 1,"},{"Start":"01:43.530 ","End":"01:45.900","Text":"so I haven\u0027t changed the exercise."},{"Start":"01:45.900 ","End":"01:49.860","Text":"Right here, the same square root,"},{"Start":"01:49.860 ","End":"01:53.670","Text":"only this time it will be plus x squared."},{"Start":"01:53.670 ","End":"01:56.370","Text":"Just copy it, x to the fourth,"},{"Start":"01:56.370 ","End":"02:03.435","Text":"plus x squared plus 1 and the difference is here\u0027s minus here\u0027s plus."},{"Start":"02:03.435 ","End":"02:06.420","Text":"But we have to write the same thing on the denominator,"},{"Start":"02:06.420 ","End":"02:08.175","Text":"so here also,"},{"Start":"02:08.175 ","End":"02:12.885","Text":"we have the square root of x to the fourth,"},{"Start":"02:12.885 ","End":"02:18.960","Text":"plus x squared plus 1 plus x squared,"},{"Start":"02:18.960 ","End":"02:20.760","Text":"so this thing is the same as this thing,"},{"Start":"02:20.760 ","End":"02:23.310","Text":"so we haven\u0027t changed the value."},{"Start":"02:23.310 ","End":"02:30.585","Text":"Now, limit as x goes to infinity,"},{"Start":"02:30.585 ","End":"02:34.215","Text":"this thing squared is just as it is."},{"Start":"02:34.215 ","End":"02:35.895","Text":"X to the fourth,"},{"Start":"02:35.895 ","End":"02:39.165","Text":"plus x squared plus 1."},{"Start":"02:39.165 ","End":"02:42.510","Text":"This thing squared minus x squared squared,"},{"Start":"02:42.510 ","End":"02:45.540","Text":"which is minus x to the fourth."},{"Start":"02:45.540 ","End":"02:50.550","Text":"That\u0027s the numerator and the denominator is what it is here,"},{"Start":"02:50.550 ","End":"02:53.880","Text":"but let\u0027s do some simplification. We\u0027ve done this before."},{"Start":"02:53.880 ","End":"03:00.510","Text":"We take x to the fourth outside the brackets and we get the square root and then we use"},{"Start":"03:00.510 ","End":"03:07.665","Text":"the formula that the square root of ab is square root of a square root of b,"},{"Start":"03:07.665 ","End":"03:10.455","Text":"and if we do this with a equals x to the fourth,"},{"Start":"03:10.455 ","End":"03:15.720","Text":"x to the fourth times 1 plus 1"},{"Start":"03:15.720 ","End":"03:21.045","Text":"over x squared plus 1 over x to the fourth,"},{"Start":"03:21.045 ","End":"03:25.095","Text":"close the bracket plus x squared."},{"Start":"03:25.095 ","End":"03:26.985","Text":"Now, we\u0027re lucky,"},{"Start":"03:26.985 ","End":"03:32.925","Text":"this thing cancels out and what we\u0027re going to do now is take"},{"Start":"03:32.925 ","End":"03:39.450","Text":"x squared outside the brackets here and that way it will cancel with this x squared,"},{"Start":"03:39.450 ","End":"03:45.225","Text":"so we get limit x goes to infinity"},{"Start":"03:45.225 ","End":"03:53.790","Text":"of x squared brackets 1 plus 1 over x squared."},{"Start":"03:53.790 ","End":"03:59.325","Text":"Here, notice that we have the square root of x to the fourth."},{"Start":"03:59.325 ","End":"04:02.715","Text":"This would normally be plus or minus x squared,"},{"Start":"04:02.715 ","End":"04:05.370","Text":"but because x is going to infinity,"},{"Start":"04:05.370 ","End":"04:08.850","Text":"then it\u0027s going to be plus x squared."},{"Start":"04:08.850 ","End":"04:11.760","Text":"The square root of x to the fourth is always x squared because"},{"Start":"04:11.760 ","End":"04:15.270","Text":"the square root has to be positive and x squared is positive,"},{"Start":"04:15.270 ","End":"04:16.905","Text":"so it has to be x squared,"},{"Start":"04:16.905 ","End":"04:19.330","Text":"can\u0027t be minus x squared."},{"Start":"04:21.140 ","End":"04:23.895","Text":"In the denominator,"},{"Start":"04:23.895 ","End":"04:27.930","Text":"this is x squared times same thing,"},{"Start":"04:27.930 ","End":"04:35.310","Text":"1 plus 1 over x squared plus 1 over x to the fourth plus x squared."},{"Start":"04:35.310 ","End":"04:38.355","Text":"If we take x squared out of the brackets here,"},{"Start":"04:38.355 ","End":"04:43.635","Text":"we\u0027ll get this thing plus 1 until the x squared basically cancel."},{"Start":"04:43.635 ","End":"04:47.250","Text":"What I\u0027m saying, is we can skip a step because we\u0027re good at algebra,"},{"Start":"04:47.250 ","End":"04:49.170","Text":"and this cancels with this,"},{"Start":"04:49.170 ","End":"04:50.250","Text":"this cancels with this,"},{"Start":"04:50.250 ","End":"04:53.080","Text":"but leaves a 1 here."},{"Start":"04:53.750 ","End":"04:58.770","Text":"What we have now is that we can now substitute x equals infinity,"},{"Start":"04:58.770 ","End":"05:00.885","Text":"and so we get, from here,"},{"Start":"05:00.885 ","End":"05:04.035","Text":"we get a 1, from here,"},{"Start":"05:04.035 ","End":"05:11.205","Text":"we get a 0 because infinity 1 over infinity is 0"},{"Start":"05:11.205 ","End":"05:21.270","Text":"divided by 1 plus 0 plus 0 plus 1."},{"Start":"05:21.270 ","End":"05:26.970","Text":"This is 1, this is 1 plus 0 plus 1 is 2,"},{"Start":"05:26.970 ","End":"05:32.860","Text":"so it\u0027s just equal to 1/2 and that\u0027s the answer."}],"ID":4777},{"Watched":false,"Name":"Exercise 23","Duration":"5m 32s","ChapterTopicVideoID":4769,"CourseChapterTopicPlaylistID":65363,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4769.jpeg","UploadDate":"2016-05-30T05:13:35.5000000","DurationForVideoObject":"PT5M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.745","Text":"In this exercise, we have to find the limit as x goes to infinity of this expression,"},{"Start":"00:05.745 ","End":"00:08.535","Text":"the square root of this minus the square root of that."},{"Start":"00:08.535 ","End":"00:11.640","Text":"We\u0027re pretty familiar with this exercise already."},{"Start":"00:11.640 ","End":"00:14.550","Text":"If we substitute x equals infinity,"},{"Start":"00:14.550 ","End":"00:18.195","Text":"we\u0027re going to get the square root of infinity minus the square root infinity."},{"Start":"00:18.195 ","End":"00:23.355","Text":"In other words, we\u0027re going to get the expression of the type infinity minus infinity."},{"Start":"00:23.355 ","End":"00:26.385","Text":"We\u0027ve already seen could be anything, it\u0027s not defined."},{"Start":"00:26.385 ","End":"00:28.620","Text":"We have to use other techniques."},{"Start":"00:28.620 ","End":"00:30.690","Text":"Square roots indicate that we should be using"},{"Start":"00:30.690 ","End":"00:35.730","Text":"conjugates and I\u0027m going to remind you what a conjugate is."},{"Start":"00:35.730 ","End":"00:39.345","Text":"If we have an expression A minus B,"},{"Start":"00:39.345 ","End":"00:45.795","Text":"its conjugate is A plus B and vice versa."},{"Start":"00:45.795 ","End":"00:50.975","Text":"We remember the formula that if we multiply these 2 conjugates,"},{"Start":"00:50.975 ","End":"00:53.475","Text":"we get difference of squares,"},{"Start":"00:53.475 ","End":"00:56.910","Text":"A squared minus B squared and this helps because if A"},{"Start":"00:56.910 ","End":"01:01.070","Text":"and/or B is square roots then squaring it gets rid of that."},{"Start":"01:01.070 ","End":"01:04.000","Text":"We\u0027re also going to need as usual,"},{"Start":"01:04.000 ","End":"01:05.690","Text":"and we use this a lot,"},{"Start":"01:05.690 ","End":"01:12.480","Text":"that a number a over infinity could be plus or minus is 0."},{"Start":"01:12.880 ","End":"01:17.030","Text":"Let\u0027s write this as the limit,"},{"Start":"01:17.030 ","End":"01:19.625","Text":"x goes to infinity."},{"Start":"01:19.625 ","End":"01:21.890","Text":"Write it as a fraction,"},{"Start":"01:21.890 ","End":"01:26.120","Text":"where here I have the original expression,"},{"Start":"01:26.120 ","End":"01:32.119","Text":"square root of x squared plus ax minus"},{"Start":"01:32.119 ","End":"01:38.295","Text":"the square root of x squared plus bx,"},{"Start":"01:38.295 ","End":"01:48.180","Text":"and then multiply it by its conjugate square root of x squared plus ax."},{"Start":"01:48.180 ","End":"01:50.715","Text":"Instead of minus, we need a plus,"},{"Start":"01:50.715 ","End":"01:53.850","Text":"plus the square root of x"},{"Start":"01:53.850 ","End":"02:01.939","Text":"squared plus bx and because we\u0027ve multiplied by the conjugate,"},{"Start":"02:01.939 ","End":"02:03.965","Text":"we also have to divide by it,"},{"Start":"02:03.965 ","End":"02:08.330","Text":"to leave our answer unchanged multiply by something over itself,"},{"Start":"02:08.330 ","End":"02:10.775","Text":"it\u0027s 1 that\u0027s okay to do."},{"Start":"02:10.775 ","End":"02:14.555","Text":"So we have here the same expression as here,"},{"Start":"02:14.555 ","End":"02:19.480","Text":"the square root of x squared plus ax,"},{"Start":"02:19.480 ","End":"02:23.060","Text":"the square root of x squared, sorry,"},{"Start":"02:23.060 ","End":"02:25.025","Text":"not minus plus same as here,"},{"Start":"02:25.025 ","End":"02:28.190","Text":"x squared plus bx."},{"Start":"02:28.190 ","End":"02:31.730","Text":"This is going to equal limit,"},{"Start":"02:31.730 ","End":"02:33.920","Text":"and using this formula,"},{"Start":"02:33.920 ","End":"02:35.660","Text":"A squared minus B squared,"},{"Start":"02:35.660 ","End":"02:41.870","Text":"we get x squared plus ax minus that\u0027s the b squared part."},{"Start":"02:41.870 ","End":"02:45.705","Text":"That\u0027s x squared plus bx."},{"Start":"02:45.705 ","End":"02:47.550","Text":"But I need brackets here,"},{"Start":"02:47.550 ","End":"02:53.119","Text":"all over square root of x squared plus ax"},{"Start":"02:53.119 ","End":"03:01.325","Text":"plus the square root of x squared plus b times x."},{"Start":"03:01.325 ","End":"03:04.490","Text":"Now, the numerator,"},{"Start":"03:04.490 ","End":"03:06.800","Text":"the x squared cancels,"},{"Start":"03:06.800 ","End":"03:12.005","Text":"and all we\u0027re left with is ax minus bx."},{"Start":"03:12.005 ","End":"03:17.120","Text":"If we do the algebra and in the denominator,"},{"Start":"03:17.120 ","End":"03:20.570","Text":"we\u0027ll take x squared outside the brackets,"},{"Start":"03:20.570 ","End":"03:30.570","Text":"so we get the square root of x squared times 1 plus a over x,"},{"Start":"03:30.570 ","End":"03:33.780","Text":"plus the square root,"},{"Start":"03:33.780 ","End":"03:41.495","Text":"and here we also take x squared outside the bracket times 1 plus b over x,"},{"Start":"03:41.495 ","End":"03:44.645","Text":"and x goes to infinity."},{"Start":"03:44.645 ","End":"03:50.510","Text":"When I point out that the square root of x squared is the absolute value of x."},{"Start":"03:50.510 ","End":"03:52.400","Text":"It could be plus or minus x."},{"Start":"03:52.400 ","End":"03:56.510","Text":"But because x is going to infinity, it\u0027s positive,"},{"Start":"03:56.510 ","End":"04:02.780","Text":"so the answer is just x and not minus x. I also want to remind you that"},{"Start":"04:02.780 ","End":"04:06.740","Text":"the square root of ab is"},{"Start":"04:06.740 ","End":"04:11.975","Text":"the square root of a times the square root of b if a and b are both positive,"},{"Start":"04:11.975 ","End":"04:15.215","Text":"and so what we\u0027re left with here is the limit,"},{"Start":"04:15.215 ","End":"04:18.450","Text":"x goes to infinity of."},{"Start":"04:18.450 ","End":"04:26.915","Text":"Now we can take x outside here and be left with a minus b, and here,"},{"Start":"04:26.915 ","End":"04:29.840","Text":"if we use this formula here to break it up,"},{"Start":"04:29.840 ","End":"04:32.690","Text":"we get square root of x"},{"Start":"04:32.690 ","End":"04:40.805","Text":"squared times square root of 1 plus a over x,"},{"Start":"04:40.805 ","End":"04:47.615","Text":"plus the square root of 1 plus b over x."},{"Start":"04:47.615 ","End":"04:50.225","Text":"Now I can close the brackets."},{"Start":"04:50.225 ","End":"04:53.780","Text":"Like we said, square root of x squared is x,"},{"Start":"04:53.780 ","End":"04:56.230","Text":"so this cancels with this,"},{"Start":"04:56.230 ","End":"05:00.110","Text":"and now we can substitute x equals infinity,"},{"Start":"05:00.110 ","End":"05:05.909","Text":"so we get a minus b"},{"Start":"05:05.909 ","End":"05:12.085","Text":"over the square root of 1 plus 0,"},{"Start":"05:12.085 ","End":"05:16.820","Text":"plus again the square root of 1 plus 0."},{"Start":"05:16.820 ","End":"05:20.810","Text":"Now the denominator, square root of 1 is 1,"},{"Start":"05:20.810 ","End":"05:22.520","Text":"1 plus 1 is 2."},{"Start":"05:22.520 ","End":"05:27.785","Text":"We\u0027re left with a minus b over 2,"},{"Start":"05:27.785 ","End":"05:31.440","Text":"and this is our answer."}],"ID":4778}],"Thumbnail":null,"ID":65363}]

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