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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 2.4.7e
Chapter 2, Problem 2.4.7e

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)

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Identify that the limit is asking for the behavior of the function \( f(x) \) as \( x \) approaches 2 from the right (denoted by \( x \to 2^+ \)).
Recognize that a vertical asymptote at \( x = 2 \) implies that as \( x \) approaches 2, \( f(x) \) tends towards infinity or negative infinity.
Since the limit is from the right, consider the values of \( f(x) \) for \( x > 2 \) and observe whether \( f(x) \) increases without bound or decreases without bound.
If the graph shows \( f(x) \) increasing as \( x \to 2^+ \), then \( \lim_{x \to 2^+} f(x) = +\infty \).
If the graph shows \( f(x) \) decreasing as \( x \to 2^+ \), then \( \lim_{x \to 2^+} f(x) = -\infty \).

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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 f has vertical asymptotes at x=1 and x=2, indicating that as x approaches these values, f(x) does not settle at a finite value but instead diverges.
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Limits

A limit describes the behavior of a function as the input approaches a particular point. The notation lim x→2^+ f(x) specifically refers to the limit of f(x) as x approaches 2 from the right side (values greater than 2). Understanding limits is crucial for analyzing the behavior of functions near points of discontinuity, such as vertical asymptotes.
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One-Sided Limits

One-Sided Limits

One-sided limits focus on the behavior of a function as the input approaches a specific point from one direction only. The notation lim x→2^+ f(x) indicates a right-hand limit, which is essential for understanding how the function behaves as it nears the vertical asymptote at x=2. This concept helps determine if the function approaches positive or negative infinity in that direction.
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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

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

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

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

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

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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