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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 2.4.7b
Chapter 2, Problem 2.4.7b

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)

Verified step by step guidance
1
Identify that the limit \( \lim_{x \to 1^+} f(x) \) involves approaching the point \( x = 1 \) from the right.
Recognize that a vertical asymptote at \( x = 1 \) implies that as \( x \) approaches 1 from the right, \( f(x) \) will tend towards either positive or negative infinity.
Examine the behavior of \( f(x) \) as \( x \to 1^+ \) by considering the values of \( f(x) \) for \( x \) slightly greater than 1.
Determine whether \( f(x) \) increases without bound (approaches \( +\infty \)) or decreases without bound (approaches \( -\infty \)) as \( x \to 1^+ \).
Conclude the analysis of the limit based on the direction in which \( f(x) \) tends as \( x \to 1^+ \).

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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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One-Sided Limits

One-sided limits refer to the behavior of a function as the input approaches a specific value from one side only. The notation lim x→1^+ f(x) indicates the limit of f(x) as x approaches 1 from the right (values greater than 1). Understanding one-sided limits is crucial for analyzing functions with vertical asymptotes.
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Limit Behavior Near Asymptotes

The limit behavior near vertical asymptotes is characterized by the function's tendency to increase or decrease without bound. For instance, if lim x→1^+ f(x) approaches positive infinity, it indicates that as x gets closer to 1 from the right, the function's values rise indefinitely. This behavior is essential for understanding the overall shape and characteristics of the graph.
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Related Practice
Textbook Question

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

The line x=−1 is a vertical asymptote of the function f(x) =x^2 − 7x + 6 / x^2 − 1.

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Textbook Question

The hyperbolic cosine function, denoted cosh(x)\(\cosh\]\left\)(x\(\right\)), is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as cosh(x)=ex+e−x2\(\cosh\]\left\)(x\(\right\))=\(\frac{e^{x}\)+e^{-x}}{2}.


b. Evaluate cosh(0)\(\cosh\]\left\)(0\(\right\)). Use symmetry and part (a) to sketch a plausible graph for y=cosh(x)y=\(\cosh\]\left\)(x\(\right\)).

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Textbook Question

Complete the following steps for the given functions. 


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


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

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Textbook Question

Let g(x)={x2+xif x<1aif x=13x+5if x>1g\(\left\)(x\(\right\))=\(\begin{cases}\)x^2+x & \(\text{if }\)x<1\\ a & \(\text{if }\)x=1\\ 3x+5 & \(\text{if }\)x>1\(\end{cases}\)

b. Determine the value of aa for which gg is continuous from the right at 11. 

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Textbook Question

Determine the following limits.


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

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Textbook Question

Use analytic methods to find the value of lim x→π/4 cos 2x / cos x − sin x.

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