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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 2.76b
Chapter 2, Problem 2.76b

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)

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To find the vertical asymptotes, set the denominator equal to zero: \(x^2 - 1 = 0\). Solve for \(x\) to find the points where the function is undefined.
This can be factored as \((x - 1)(x + 1) = 0\). Therefore, the solutions are \(x = 1\) and \(x = -1\). These are the potential vertical asymptotes.
Check if the numerator \(x^4 - 1\) can be factored to cancel any common factors with the denominator. Factor \(x^4 - 1\) as \((x^2 - 1)(x^2 + 1)\).
The expression becomes \(\frac{(x^2 - 1)(x^2 + 1)}{x^2 - 1}\). Cancel the common factor \(x^2 - 1\), leaving \(f(x) = x^2 + 1\) for \(x \neq \pm 1\).
For \(x = 1\) and \(x = -1\), analyze \(\lim_{x \to 1^-} f(x)\), \(\lim_{x \to 1^+} f(x)\), \(\lim_{x \to -1^-} f(x)\), and \(\lim_{x \to -1^+} f(x)\) using the simplified function \(f(x) = x^2 + 1\).

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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 a function when the output approaches infinity as the input approaches a certain value from either the left or the right. This typically happens at points where the function is undefined, often due to division by zero. Identifying vertical asymptotes involves finding the values of x that make the denominator zero while ensuring the numerator is not also zero at those points.
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Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In the context of vertical asymptotes, we analyze the left-hand limit (as x approaches a from the left) and the right-hand limit (as x approaches a from the right) to determine the behavior of the function near the asymptote. This helps in understanding whether the function tends to positive or negative infinity.
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Factoring Polynomials

Factoring polynomials is a technique used to simplify expressions, making it easier to analyze their behavior, such as finding asymptotes. In the given function f(x) = (x^4−1)/(x^2−1), factoring both the numerator and denominator can reveal common factors and help identify points of discontinuity. This process is crucial for determining where vertical asymptotes may exist and for simplifying the function before evaluating limits.
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Related Practice
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

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

Tangent lines with zero slope


a. Graph the function f(x)=x^2−4x+3.

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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)

Textbook Question

Let g(x)=x3−4x8∣x−2∣g\(\left\)(x\(\right\))=\(\frac{x^3-4x}{8\left|x-2\right|}\). <IMAGE>

Calculate g(x)g\(\left\)(x\(\right\)) for each value of xx in the following table.