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(a) On the graph above, label the x and y-intercepts, the holes and the vertical asymptotes. Be sure to use equations of lines for the vertical asymptotes. (b) Write the limits which describe the end-behavior of the graph. (c) For each vertical asymptote, write down the two limits that the describe the graph of the function near that asymptote.

(a) On the graph above, label the x and y-intercepts, the holes and the vertical asymptotes. Be sure to use equations of lines for the vertical asymptotes. (b) Write the limits which describe the end-behavior of the graph. (c) For each vertical asymptote, write down the two limits that the describe the graph of the function near that asymptote.

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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### Graph Analysis and Explanation #### Graph Description: The graph is a plot of a function with notable features such as vertical asymptotes and a hole. It shows two separate branches, indicating a rational function or something similar. - **Axes:** The graph is on a Cartesian coordinate plane with both x and y axes marked at intervals of 1 unit. - **Vertical Asymptotes:** The graph has vertical asymptotes at \( x = 0 \) and \( x = 2 \). These are represented by the branches approaching the lines but never touching them. - **Hole:** There is a hole at \( (6, 0) \), indicated by an empty circle. - **Endpoints:** Both left and right ends of the graph appear to approach horizontal asymptotes. #### Instructions for Analysis: (a) **Labeling Intercepts, Holes, and Asymptotes:** - **x-intercepts:** There is one x-intercept at the origin where the graph crosses the x-axis. - **y-intercepts:** The y-intercept is at the origin as well. - **Holes:** There is a hole at \( (6, 0) \). - **Vertical Asymptotes:** As stated, they exist at \( x = 0 \) and \( x = 2 \). (b) **End-Behavior Limits:** - As \( x \to -\infty \), the graph levels off towards a horizontal line, suggesting a horizontal asymptote. - As \( x \to \infty \), similarly, the graph levels off towards the same horizontal line. (c) **Vertical Asymptote Limits:** - As \( x \to 0^+ \) and \( x \to 0^- \), the graph approaches \(\pm \infty\) respectively. - As \( x \to 2^+ \) and \( x \to 2^- \), the graph also approaches \(\pm \infty \). This graph represents a function with distinct features important for understanding limits and asymptotic behavior.
Transcribed Image Text:### Graph Analysis and Explanation #### Graph Description: The graph is a plot of a function with notable features such as vertical asymptotes and a hole. It shows two separate branches, indicating a rational function or something similar. - **Axes:** The graph is on a Cartesian coordinate plane with both x and y axes marked at intervals of 1 unit. - **Vertical Asymptotes:** The graph has vertical asymptotes at \( x = 0 \) and \( x = 2 \). These are represented by the branches approaching the lines but never touching them. - **Hole:** There is a hole at \( (6, 0) \), indicated by an empty circle. - **Endpoints:** Both left and right ends of the graph appear to approach horizontal asymptotes. #### Instructions for Analysis: (a) **Labeling Intercepts, Holes, and Asymptotes:** - **x-intercepts:** There is one x-intercept at the origin where the graph crosses the x-axis. - **y-intercepts:** The y-intercept is at the origin as well. - **Holes:** There is a hole at \( (6, 0) \). - **Vertical Asymptotes:** As stated, they exist at \( x = 0 \) and \( x = 2 \). (b) **End-Behavior Limits:** - As \( x \to -\infty \), the graph levels off towards a horizontal line, suggesting a horizontal asymptote. - As \( x \to \infty \), similarly, the graph levels off towards the same horizontal line. (c) **Vertical Asymptote Limits:** - As \( x \to 0^+ \) and \( x \to 0^- \), the graph approaches \(\pm \infty\) respectively. - As \( x \to 2^+ \) and \( x \to 2^- \), the graph also approaches \(\pm \infty \). This graph represents a function with distinct features important for understanding limits and asymptotic behavior.
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