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does the limit not exist

does the limit not exist

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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why does the limit not exist without mentioning derivative not existing

### Understanding Limits and Differentiability #### Limits and Non-Differentiability As we know, limits do not always exist. Here are some examples of functions that are not differentiable at the point \( x = 1 \). For each, explain why the limit below does not exist. Avoid reasoning by claiming the derivative doesn’t exist, as that would be a circular argument! \[ \lim_{{x \to a}} \frac{{f(x) - f(a)}}{{x - a}} \] #### Graph Analysis **Graph (a):** - The graph presents a curve that exhibits a sharp peak around \( x = 1 \). - The function is smooth from \( x = -1 \) to just before \( x = 1 \) and from just after \( x = 1 \) to \( x = 2.5 \). - At \( x = 1 \), the graph has a clear angular point, indicating a cusp or a point of non-smoothness. - This non-smooth point implies that the slope of the tangent line cannot be consistently defined at \( x = 1 \), hence the limit does not exist at that point. To fully understand why the limit fails to exist, consider how the slope of the secant lines behaves as they approach \( x = 1 \) from either side. This inconsistency in approaching the slope confirms non-differentiability at \( x = 1 \).
Transcribed Image Text:### Understanding Limits and Differentiability #### Limits and Non-Differentiability As we know, limits do not always exist. Here are some examples of functions that are not differentiable at the point \( x = 1 \). For each, explain why the limit below does not exist. Avoid reasoning by claiming the derivative doesn’t exist, as that would be a circular argument! \[ \lim_{{x \to a}} \frac{{f(x) - f(a)}}{{x - a}} \] #### Graph Analysis **Graph (a):** - The graph presents a curve that exhibits a sharp peak around \( x = 1 \). - The function is smooth from \( x = -1 \) to just before \( x = 1 \) and from just after \( x = 1 \) to \( x = 2.5 \). - At \( x = 1 \), the graph has a clear angular point, indicating a cusp or a point of non-smoothness. - This non-smooth point implies that the slope of the tangent line cannot be consistently defined at \( x = 1 \), hence the limit does not exist at that point. To fully understand why the limit fails to exist, consider how the slope of the secant lines behaves as they approach \( x = 1 \) from either side. This inconsistency in approaching the slope confirms non-differentiability at \( x = 1 \).
The image displays a graph labeled "(b)" that appears to represent a sigmoid curve. ### Graph Description: - **Axes**: - The horizontal axis (x-axis) ranges approximately from -2 to 3. - The vertical axis (y-axis) ranges from -1 to 1.5. - **Curve**: - The curve starts at the bottom left, with a y-value around -0.5 as the x-value increases from -2. - It smoothly ascends through the origin, where it crosses the y-axis. - The curve continues to rise, approaching a y-value near 1 as the x-value approaches 3. ### Explanation: This graph is characteristic of a sigmoid function, often used in logistic regression and neural networks as an activation function due to its property of mapping any real-valued number into a range between 0 and 1. The sigmoid function is useful in probability estimation, where outcomes are modeled as binary. This type of graph is particularly important in the fields of machine learning and statistics for modeling growth and saturation processes.
Transcribed Image Text:The image displays a graph labeled "(b)" that appears to represent a sigmoid curve. ### Graph Description: - **Axes**: - The horizontal axis (x-axis) ranges approximately from -2 to 3. - The vertical axis (y-axis) ranges from -1 to 1.5. - **Curve**: - The curve starts at the bottom left, with a y-value around -0.5 as the x-value increases from -2. - It smoothly ascends through the origin, where it crosses the y-axis. - The curve continues to rise, approaching a y-value near 1 as the x-value approaches 3. ### Explanation: This graph is characteristic of a sigmoid function, often used in logistic regression and neural networks as an activation function due to its property of mapping any real-valued number into a range between 0 and 1. The sigmoid function is useful in probability estimation, where outcomes are modeled as binary. This type of graph is particularly important in the fields of machine learning and statistics for modeling growth and saturation processes.
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