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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 2.4.7c
Chapter 2, Problem 2.4.7c

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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Identify the behavior of the function \( f(x) \) as \( x \) approaches 1 from the left (\( x \to 1^- \)).
Identify the behavior of the function \( f(x) \) as \( x \) approaches 1 from the right (\( x \to 1^+ \)).
Determine if the left-hand limit and the right-hand limit as \( x \to 1 \) are equal or not.
If the left-hand limit and right-hand limit are not equal, conclude that the limit \( \lim_{x \to 1} f(x) \) does not exist.
If the left-hand limit and right-hand limit are equal, state that the limit \( \lim_{x \to 1} f(x) \) is equal to that common value.

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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 limit but instead diverges.
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Limits

A limit describes the behavior of a function as the input approaches a particular value. In the context of vertical asymptotes, the limit of f(x) as x approaches 1 or 2 will help determine whether f(x) approaches positive or negative infinity, which is crucial for understanding the function's behavior near these points.
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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 the left or the right. For the limits as x approaches 1, it is important to evaluate both the left-hand limit (as x approaches 1 from values less than 1) and the right-hand limit (as x approaches 1 from values greater than 1) to fully understand the function's behavior at the asymptote.
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One-Sided Limits
Related Practice
Textbook Question

Complete the following steps for the given functions. 


c. Graph ff 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)=x2−3x+6f\(\left\)(x\(\right\))=\(\frac{x^2-3}{x+6}\)

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

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 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)?