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Given the provided graph of f(x), find the following limits. a) As the limit x nears (-infinity): f(x)= b) As the limit x nears (infinity): f(x)=

Given the provided graph of f(x), find the following limits. a) As the limit x nears (-infinity): f(x)= b) As the limit x nears (infinity): f(x)=

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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Given the provided graph of f(x), find the following limits.

a) As the limit x nears (-infinity): f(x)=

b) As the limit x nears (infinity): f(x)=

 

The image features a graph of the function \( y = \frac{1}{x^2} \). ### Description of the Graph: - **Axes and Grid**: The graph is plotted on a coordinate plane with both the x-axis and y-axis labeled. The x-axis ranges from -6 to 6, and the y-axis ranges from -6 to 6. The gridlines are evenly spaced, providing a clear reference for values. - **Function Behavior**: - As \( x \) approaches 0 from either direction, the function \( y = \frac{1}{x^2} \) increases towards positive infinity. This is depicted by the graph's sharp upward direction near the y-axis. - The curve is symmetrical about the y-axis, indicating that the function is even. - For large absolute values of \( x \) (both positive and negative), the function's value approaches 0, which is shown by the graph flattening as it extends outward. - **Key Features**: - There are vertical asymptotes at \( x = 0 \), meaning the graph gets infinitely close to the y-axis but never touches or intersects it. - There are no x-intercepts or real y-intercepts for this graph since the function does not equal zero or attain a finite value on the y-axis. This graph effectively visualizes the characteristics of the rational function \( \frac{1}{x^2} \), highlighting its asymptotic behavior and symmetry.
Transcribed Image Text:The image features a graph of the function \( y = \frac{1}{x^2} \). ### Description of the Graph: - **Axes and Grid**: The graph is plotted on a coordinate plane with both the x-axis and y-axis labeled. The x-axis ranges from -6 to 6, and the y-axis ranges from -6 to 6. The gridlines are evenly spaced, providing a clear reference for values. - **Function Behavior**: - As \( x \) approaches 0 from either direction, the function \( y = \frac{1}{x^2} \) increases towards positive infinity. This is depicted by the graph's sharp upward direction near the y-axis. - The curve is symmetrical about the y-axis, indicating that the function is even. - For large absolute values of \( x \) (both positive and negative), the function's value approaches 0, which is shown by the graph flattening as it extends outward. - **Key Features**: - There are vertical asymptotes at \( x = 0 \), meaning the graph gets infinitely close to the y-axis but never touches or intersects it. - There are no x-intercepts or real y-intercepts for this graph since the function does not equal zero or attain a finite value on the y-axis. This graph effectively visualizes the characteristics of the rational function \( \frac{1}{x^2} \), highlighting its asymptotic behavior and symmetry.
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