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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 2.4.9f
Chapter 2, Problem 2.4.9f

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

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Step 1: Identify the type of limit problem. Since the limit is approaching a point where there is a vertical asymptote (x = 3), this is a limit involving a vertical asymptote.
Step 2: Understand the behavior of the function near the asymptote. As x approaches 3, the function h(x) will tend to either positive infinity or negative infinity, depending on the direction from which x approaches 3.
Step 3: Consider the one-sided limits. Evaluate the limit from the left (x → 3⁻) and from the right (x → 3⁺) separately to determine the behavior of h(x) as it approaches the asymptote.
Step 4: Analyze the graph. Look at the graph of h(x) near x = 3 to see if the function approaches positive or negative infinity from either side.
Step 5: Conclude the limit. Based on the behavior observed in the graph, determine if the two one-sided limits are equal or if they differ, which will indicate whether the overall limit exists or not.

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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, h(x) will not approach a finite limit.
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Limits

A limit describes the behavior of a function as the input approaches a particular value. In the context of the question, evaluating the limit as x approaches 3 for h(x) involves determining how h(x) behaves near the vertical asymptote at x = 3, which typically results in the limit being either positive or negative infinity.
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One-Sided Limits

One-Sided Limits

One-sided limits are used to analyze the behavior of a function as it approaches a specific point from one side only, either from the left (denoted as lim x→c-) or from the right (denoted as lim x→c+). For the limit as x approaches 3, it is essential to consider both one-sided limits to fully understand the behavior of h(x) near the vertical asymptote.
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Related Practice
Textbook Question

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

lim x→3^+ h(x)

Textbook Question

What does it mean for a function to be continuous on an interval?

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Textbook Question

The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>

lim x→2^+ f(x)

Textbook Question

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


d. For what values of t on the interval [0, 9] is the instantaneous velocity positive (the projectile moves upward)?

Textbook Question

21–30. Derivatives

a. Use limits to find the derivative function f' for the following functions f.

f(x) = 5x+2; a=1, 2

Textbook Question

The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>

lim x→2 f (x)