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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 2.4.7d
Chapter 2, Problem 2.4.7d

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 approaching from the left side of x = 2, denoted as x \(\to\) 2^-.
Recognize that the function f(x) has a vertical asymptote at x = 2, which means the function's values increase or decrease without bound as x approaches 2.
Since the limit is from the left, consider the behavior of f(x) as x approaches 2 from values less than 2.
Analyze the graph of f(x) to determine whether the function approaches positive infinity or negative infinity as x approaches 2 from the left.
Conclude that the limit \( \lim_{x \to 2^-} f(x) \) is either \( +\infty \) or \( -\infty \) based on the observed behavior of the graph.

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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 left side. 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 evaluate the behavior of a function as the input approaches a specific point from one direction only. The left-hand limit, denoted as lim x→2^− f(x), examines the values of f(x) as x approaches 2 from values less than 2. This concept is essential for understanding how functions behave near vertical asymptotes.
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One-Sided Limits
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)=x2−2x+53x−2f\(\left\)(x\(\right\))=\(\frac{x^2-2x+5}{3x-2}\)

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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→1 f(x)

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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

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

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)=3x2−2x+53x+4f\(\left\)(x\(\right\))=\(\frac{3x^2-2x+5}{3x+4}\)

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)?