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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)

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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

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a- f(x) and lim x→a+ f(x).

f(x) = (x^4−1)/(x^2−1)

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\(\right\)).

Textbook Question

Assume you invest \(250 at the end of each year for 10 years at an annual interest rate of rr. The amount of money in your account after 10 years is given by A(r)=250((1+r)10−1)rA\left(r\right)=\frac{250\left(\left(1+r\right)^{10}-1\right)}{r}. Assume your goal is to have \)3500 in your account after 10 years.


b. Use a calculator to estimate the interest rate required to reach your financial goal.

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

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.


b. From the graph of the position function, identify the time at which the projectile has an instantaneous velocity of zero; call this time t=a.