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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 2.4.9c
Chapter 2, Problem 2.4.9c

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: Identify the type of limit problem. Since the problem involves a vertical asymptote at x = -2, we are dealing with a limit where the function approaches infinity or negative infinity.
Step 2: Understand the behavior of the function near the asymptote. As x approaches -2, the function h(x) will either increase or decrease without bound.
Step 3: Consider the direction of approach. Determine if you need to evaluate the limit from the left (x approaches -2 from the left, denoted as x → -2⁻) or from the right (x approaches -2 from the right, denoted as x → -2⁺).
Step 4: Analyze the graph of h(x) near x = -2. Observe whether the function values are increasing towards positive infinity or decreasing towards negative infinity as x approaches -2 from either side.
Step 5: Conclude the limit based on the behavior observed. If the function approaches positive infinity from both sides, the limit is positive infinity. If it approaches negative infinity, the limit is negative infinity. If the behavior differs from each side, the limit does not exist.

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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(x) has vertical asymptotes at x = -2 and x = 3, indicating that as x approaches these values, h(x) will either increase or decrease without bound.
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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 of h(x) as x approaches -2 involves determining what value h(x) approaches as x gets closer to -2, which is critical for understanding the function's behavior near its vertical asymptote.
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One-Sided Limits

One-sided limits refer to the limits of a function as the input approaches a specific value from one side only, either the left or the right. For the limit lim x→−2 h(x), it is important to consider both the left-hand limit (as x approaches -2 from values less than -2) and the right-hand limit (as x approaches -2 from values greater than -2) to fully understand the behavior of h(x) near the asymptote.
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Related Practice
Textbook Question

For the following position functions, make a table of average velocities similar to those in Exercises 19–20 and make a conjecture about the instantaneous velocity at the indicated time. 


c. s(t)=40 sin 2t at t=0

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

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

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Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.


c. The graph of a function can have any number of vertical asymptotes but at most two horizontal asymptotes.

Textbook Question

Complete the following steps for the given functions. 


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


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

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Textbook Question

Determine the following limits.


b. limx→2x−2x5−4x3{\(\displaystyle\]\lim\)_{x\(\to\)2}}\(\frac{x-2}{x^5-4x^3}\)