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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 78a
Chapter 2, Problem 78a

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.


f(x)=|1−x^2| / x(x+1)

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The function given is \( f(x) = \frac{|1-x^2|}{x(x+1)} \). We need to analyze the behavior of this function as \( x \to \infty \) and \( x \to -\infty \).
For large values of \( x \), the term \( x^2 \) dominates \( 1 \) in \( |1-x^2| \), so \( |1-x^2| \approx |x^2| = x^2 \). Thus, \( f(x) \approx \frac{x^2}{x(x+1)} = \frac{x^2}{x^2 + x} \).
Simplify \( \frac{x^2}{x^2 + x} \) by dividing the numerator and the denominator by \( x^2 \), resulting in \( \frac{1}{1 + \frac{1}{x}} \). As \( x \to \infty \), \( \frac{1}{x} \to 0 \), so \( \lim_{x \to \infty} f(x) = \frac{1}{1+0} = 1 \).
For \( x \to -\infty \), the simplification is similar: \( f(x) \approx \frac{x^2}{x^2 + x} = \frac{1}{1 + \frac{1}{x}} \). As \( x \to -\infty \), \( \frac{1}{x} \to 0 \), so \( \lim_{x \to -\infty} f(x) = 1 \).
Since both \( \lim_{x \to \infty} f(x) \) and \( \lim_{x \to -\infty} f(x) \) are equal to 1, the horizontal asymptote of the function is \( y = 1 \).

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Key Concepts

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

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the input approaches positive or negative infinity. This analysis helps determine how the function behaves in extreme cases, which is crucial for identifying horizontal asymptotes. For example, if the limit of f(x) as x approaches infinity exists and is a finite number, it indicates a horizontal asymptote at that value.
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Horizontal Asymptotes

Horizontal asymptotes are lines that a graph approaches as x approaches infinity or negative infinity. They represent the limiting value of a function in these extreme cases. To find horizontal asymptotes, one typically evaluates the limits of the function at both ends of the x-axis, determining if the function stabilizes at a particular value.
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Graphs of Exponential Functions

Absolute Value Functions

Absolute value functions, such as |1 - x^2|, affect the behavior of a function by ensuring that outputs are non-negative. This can change the function's limits and asymptotic behavior, especially when the expression inside the absolute value changes sign. Understanding how to handle absolute values is essential for accurately analyzing limits and identifying asymptotes.
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Initial Value Problems
Related Practice
Textbook Question

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Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.


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Explain why lim x→3^+ √ x−3 / 2−x does not exist.

Textbook Question

A right circular cylinder with a height of 10 cm and a surface area of S cm2 has a radius given by r(S)=1/2(√100+2S/π −10).

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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)=|1−x^2| / x(x+1)