Statistics

Biochemistry

Calculus I, II and III

To address the problem, we will analyze the behavior of a function \( f \) defined on an interval \( (a, b) \), where \( a \) and \( b \) can potentially be \( -\infty \) and \( \infty \), respectively. We will explore the limits of \( f \) as \( x \) approaches \( a \) and \( b \) from within the interval, under the conditions that \( f \) is either increasing or decreasing. This analysis will help us understand the asymptotic behavior of \( f \) near the boundaries of its domain. ### Part (a): \( f \) is an Increasing Function 1. **Understanding the Behavior of Increasing Functions**: An increasing function is one where if \( x_1 < x_2 \), then \( f(x_1) \leq f(x_2) \). This implies that as \( x \) increases, \( f(x) \) does not decrease. 2. **Limit as \( x \) Approaches \( b^- \)**: - Since \( f \) is increasing, \( f(x) \) will reach its highest value as \( x \) approaches \( b \) from the left. - The supremum (least upper bound) of \( f(x) \) over the interval \( (a, b) \) represents the maximum limit that \( f(x) \) approaches but does not necessarily attain. - Therefore, we have: \[ \lim_{x \to b^-} f(x) = \sup\{f(x) : x \in (a, b)\} \] 3. **Limit as \( x \) Approaches \( a^+ \)**: - As \( x \) increases from \( a \), \( f(x) \) starts from its lowest value within the interval. - The infimum (greatest lower bound) of \( f(x) \) over the interval \( (a, b) \) represents the minimum limit that \( f(x) \) approaches as \( x \) increases from \( a \). - Thus, we can write: \[ \lim_{x \to a^+} f(x) = \inf\{f(x) : x \in (a, b)\} \] ### Part (b): \( f \) is a Decreasing Function 1. **Understanding the Behavior of Decreasing Functions**: A decreasing function is one where if \( x_1 < x_2 \), then \( f(x_1) \geq f(x_2) \). This implies that as \( x \) increases, \( f(x) \) does not increase. 2. **Limit as \( x \) Approaches \( b^- \)**: - Since \( f \) is decreasing, \( f(x) \) will reach its lowest value as \( x \) approaches \( b \) from the left. - The infimum of \( f(x) \) over the interval \( (a, b) \) represents the minimum limit that \( f(x) \) approaches but does not necessarily attain. - Therefore, we have: \[ \lim_{x \to b^-} f(x) = \inf\{f(x) : x \in (a, b)\} \] 3. **Limit as \( x \) Approaches \( a^+ \)**: - As \( x \) increases from \( a \), \( f(x) \) starts from its highest value within the interval. - The supremum of \( f(x) \) over the interval \( (a, b) \) represents the maximum limit that \( f(x) \) approaches as \( x \) increases from \( a \). - Thus, we can write: \[ \lim_{x \to a^+} f(x) = \sup\{f(x) : x \in (a, b)\} \] This detailed analysis provides a clear understanding of how the limits of an increasing or decreasing function behave near the boundaries of its domain, which is crucial for various applications in mathematical analysis and calculus.

Math

Let \( f \) be defined on an interval \( (a, b) \), where \( a = -\infty \) and \( b = \infty \) are allowed. With the convenient notation \( a^+ = -\infty \) if \( a = -\infty \) and \( b^- = \infty \) if \( b = \infty \), we obtain (a) if \( f \) is increasing, then \[ \lim_{x \to b^-} f(x) = \sup\{f(x) : x \in (a, b)\} \quad \text{and} \quad \lim_{x \to a^+} f(x) = \inf\{f(x) : x \in (a, b)\}; \] (b) if \( f \) is decreasing, then \[ \lim_{x \to b^-} f(x) = \inf\{f(x) : x \in (a, b)\} \quad \text{and} \quad \lim_{x \to a^+} f(x) = \sup\{f(x) : x \in (a, b)\}. \]

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