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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 2.4.53b
Chapter 2, Problem 2.4.53b

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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Step 1: Identify the function f(x) = \(\frac{x^2 - 7x + 6}{x^2 - 1}\). This is a rational function, which means it is a fraction where both the numerator and the denominator are polynomials.
Step 2: Determine the points where the denominator is zero, as these are potential vertical asymptotes. Set the denominator equal to zero: x^2 - 1 = 0.
Step 3: Solve the equation x^2 - 1 = 0. This can be factored as (x - 1)(x + 1) = 0, giving the solutions x = 1 and x = -1.
Step 4: Check if x = -1 is a vertical asymptote by ensuring it is not a removable discontinuity. Factor the numerator: x^2 - 7x + 6 = (x - 1)(x - 6).
Step 5: Since x = -1 does not cancel out with any factor in the numerator, it is indeed a vertical asymptote. Therefore, the statement is true.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vertical Asymptotes

A vertical asymptote occurs in a function when the function approaches infinity or negative infinity as the input approaches a certain value. This typically happens at values of x that make the denominator of a rational function equal to zero, provided that the numerator does not also equal zero at that point.
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Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. The general form is f(x) = P(x)/Q(x), where P and Q are polynomials. Understanding the behavior of rational functions, especially their asymptotic behavior, is crucial for analyzing their graphs and determining points of discontinuity.
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Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors, which can simplify the analysis of functions. For example, the function f(x) = (x^2 - 7x + 6)/(x^2 - 1) can be factored to identify points where the function is undefined, helping to determine the presence of vertical asymptotes.
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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

Complete the following steps for the given functions. 


a. Find the slant asymptote of ff.


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

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

Tangent lines with zero slope


a. Graph the function f(x)=x^2−4x+3.

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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

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)