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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)

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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.

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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.
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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.
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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.
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Related Practice
Textbook Question

Determine the following limits.


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

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Textbook Question

Complete the following steps for the given functions. 


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


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

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\(\right\))=\(\frac{x^2-3}{x+6}\)

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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

Use analytic methods to find the value of lim x→π/4 cos 2x / cos x − sin x.

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Textbook Question

Determine the following limits.


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