Calculus: Early Transcendentals, 8th Edition
Authors: James Stewart
ISBN-13: 978-1285741550
See our solution for Question 7E from Chapter 2.6 from Stewart's Calculus, 8th Edition.
Problem 7E
Chapter:
Problem:
Sketch the graph of an example of a function f that satisfies all of the given conditions.
Step-by-Step Solution
Given information
We are given with following condition for the function\[\begin{array}{l}\mathop {\lim }\limits_{x \to 2} f\left( x \right) = - \infty ,\,\,\mathop {\lim }\limits_{x \to \infty } f\left( x \right) = \infty ,\,\,\,\mathop {\lim }\limits_{x \to - - \infty } f\left( x \right) = 0,\,\,\\\\\mathop {\lim }\limits_{x \to 0 + } f\left( x \right) = \infty ,\,\,\mathop {\lim }\limits_{x \to {0^ - }} f\left( x \right) = - \infty ,\end{array}\]
Step 1:
The function should have following properties: (1) At x=2, the function approaches negative infinity. So this is one of the asymptote
(2) At x=0, the function approaches negative infinity from left and positive infinity from right. So x=0 is another asymptote.
(3) As x reaches infinity, function also reaches infinity. Hence y=x is one of the asymptote.
(4) As x reaches negative infinity, function reaches 0. Hence y=0 is one of the asymptote.
The plot of the function is:
https://imgur.com/6AKCXgF
We are given with following condition for the function\[\begin{array}{l}\mathop {\lim }\limits_{x \to 2} f\left( x \right) = - \infty ,\,\,\mathop {\lim }\limits_{x \to \infty } f\left( x \right) = \infty ,\,\,\,\mathop {\lim }\limits_{x \to - - \infty } f\left( x \right) = 0,\,\,\\\\\mathop {\lim }\limits_{x \to 0 + } f\left( x \right) = \infty ,\,\,\mathop {\lim }\limits_{x \to {0^ - }} f\left( x \right) = - \infty ,\end{array}\]
Step 1:
The function should have following properties: (1) At x=2, the function approaches negative infinity. So this is one of the asymptote
(2) At x=0, the function approaches negative infinity from left and positive infinity from right. So x=0 is another asymptote.
(3) As x reaches infinity, function also reaches infinity. Hence y=x is one of the asymptote.
(4) As x reaches negative infinity, function reaches 0. Hence y=0 is one of the asymptote.
The plot of the function is:
https://imgur.com/6AKCXgF