Skip to main content
Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.12b
Chapter 2, Problem 2.12b

Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.


b. lim x→−2 f(x)

Verified step by step guidance
1
Identify the function: \( f(x) = \frac{e^{-x}}{x(x+2)^2} \).
Recognize that the limit \( \lim_{x \to -2} f(x) \) involves a point where the denominator becomes zero, indicating a potential vertical asymptote or removable discontinuity.
Analyze the behavior of the function as \( x \) approaches \(-2\) from both the left and the right to determine if the limit exists.
Consider the sign and magnitude of the numerator \( e^{-x} \) and the denominator \( x(x+2)^2 \) as \( x \to -2^- \) and \( x \to -2^+ \).
Use a graphing utility to visualize the function near \( x = -2 \) to confirm the behavior and determine if the limit exists or if there is a vertical asymptote.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding the function's behavior near points of interest, including points of discontinuity or where the function may not be explicitly defined. Evaluating limits is crucial for determining the continuity and differentiability of functions.
Recommended video:
05:50
One-Sided Limits

Graphing Functions

Graphing functions involves plotting the values of a function on a coordinate system to visualize its behavior. This process can reveal important features such as intercepts, asymptotes, and intervals of increase or decrease. Using a graphing utility allows for experimentation with different viewing windows, which can help in identifying the limits and overall shape of the function.
Recommended video:
5:53
Graph of Sine and Cosine Function

Asymptotic Behavior

Asymptotic behavior refers to how a function behaves as it approaches a certain point, particularly at infinity or near points of discontinuity. In the context of limits, understanding asymptotic behavior is essential for determining the value of a limit as the input approaches a specific value, such as -2 in this case. It often involves analyzing the function's growth rates and identifying any vertical or horizontal asymptotes.
Recommended video:
03:07
Cases Where Limits Do Not Exist
Related Practice
Textbook Question

Complete the following steps for the given functions. 


c. Graph f and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph.


f(x)=4x3+4x2+7x+4x2+1f\(\left\)(x\(\right\))=\(\frac{4x^3+4x^2+7x+4}{x^2+1}\)

5
views
Textbook Question

Complete the following steps for the given functions. 


b. Find the vertical asymptotes of ff (if any).


f(x)=x2−3x+6f\(\left\)(x\(\right\))=\(\frac{x^2-3}{x+6}\)

2
views
Textbook Question

Complete the following steps for the given functions. 


b. Find the vertical asymptotes of ff (if any).


f(x)=x2−2x+53x−2f\(\left\)(x\(\right\))=\(\frac{x^2-2x+5}{3x-2}\)

2
views
Textbook Question

Determine the following limits.


b. limx→2x−2x5−4x3{\(\displaystyle\]\lim\)_{x\(\to\)2}}\(\frac{x-2}{x^5-4x^3}\)

Textbook Question

Use the graph of gg in the figure to find the following values or state that they do not exist. <IMAGE>

limx→0g(x){\(\displaystyle\]\lim\)_{x\(\to\)0}g\(\left\)(x\(\right\))}

Textbook Question

Determine the following limits.


b. limx→3x−3x4−9x2{\(\displaystyle\]\lim\)_{x\(\to\)3}}\(\frac{x-3}{x^4-9x^2}\)