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 2.4.9b
Chapter 2, Problem 2.4.9b

The graph of ℎ in the figure has vertical asymptotes at x=−2 and x=3. Analyze the following limits. <IMAGE>
lim x→−2^+ h(x)

Verified step by step guidance
1
Step 1: Understand the concept of a vertical asymptote. A vertical asymptote at x = a means that as x approaches a, the function h(x) tends to infinity or negative infinity.
Step 2: Identify the direction of approach. The limit x → -2^+ indicates that we are approaching x = -2 from the right side (values greater than -2).
Step 3: Analyze the behavior of h(x) as x approaches -2 from the right. Since there is a vertical asymptote at x = -2, observe whether h(x) increases towards positive infinity or decreases towards negative infinity.
Step 4: Use the graph to determine the behavior. Look at the graph of h(x) near x = -2 from the right side to see if the function is going upwards or downwards.
Step 5: Conclude the limit based on the observed behavior. If h(x) goes to positive infinity, the limit is positive infinity. If it goes to negative infinity, the limit is negative infinity.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

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

Vertical Asymptotes

Vertical asymptotes occur in the graph of a function where the function approaches infinity or negative infinity as the input approaches a certain value. In this case, the function h has vertical asymptotes at x = -2 and x = 3, indicating that as x approaches these values, the function's output becomes unbounded.
Recommended video:
3:40
Introduction to Cotangent Graph Example 1

One-Sided Limits

One-sided limits evaluate the behavior of a function as the input approaches a specific value from one side only. The notation lim x→−2^+ h(x) indicates that we are interested in the limit of h(x) as x approaches -2 from the right (values greater than -2), which helps in understanding the function's behavior near the asymptote.
Recommended video:
05:50
One-Sided Limits

Limit Behavior Near Asymptotes

The behavior of limits near vertical asymptotes is crucial for understanding how functions behave at points where they are undefined. As x approaches a vertical asymptote, the function typically tends to either positive or negative infinity, which can be determined by analyzing the function's values just before and after the asymptote.
Recommended video:
03:07
Cases Where Limits Do Not Exist
Related Practice
Textbook Question

Determine the following limits.


b. limx→2−1x(x−2){\(\displaystyle\]\lim\)_{x\(\to\)2^{-}}}\(\frac{1}{\sqrt{x\left(x-2\right)}\)}

407
views
Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.

The line x=−1 is a vertical asymptote of the function f(x) =x^2 − 7x + 6 / x^2 − 1.

281
views
Textbook Question

Determine the following limits.


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

319
views
Textbook Question

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.


b. From the graph of the position function, identify the time at which the projectile has an instantaneous velocity of zero; call this time t=a.

463
views
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\(\right\))}

389
views
Textbook Question

Determine the following limits.


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

322
views