Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 2 - Section 2.6 - Limits at Infinity; Horizontal Asymptotes - 2.6 Exercises - Page 138: 44

Answer

(a) $\lim\limits_{x \to \infty}f(x) = 0$ (b) $\lim\limits_{x \to 0^+}f(x) = \infty$ (c) $\lim\limits_{x \to 1^-}f(x) = \infty$ (d) $\lim\limits_{x \to 1^-}f(x) = -\infty$ (e) We can see a sketch of the graph below.
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Work Step by Step

$f(x) = \frac{2}{x} - \frac{1}{ln~x}$ (a) As $~~x \to \infty~~$, the value of $\frac{2}{x}$ approaches $0$ while $\frac{1}{ln~x}$ approaches $0$ $\lim\limits_{x \to \infty}f(x) = 0$ (b) As $~~x \to 0^+~~$, the value of $\frac{2}{x}$ approaches $\infty$ while $\frac{1}{ln~x}$ approaches $0$ $\lim\limits_{x \to 0^+}f(x) = \infty$ (c) As $~~x \to 1^-~~$, the value of $\frac{2}{x}$ approaches $2$ while $\frac{1}{ln~x}$ approaches $-\infty$ $\lim\limits_{x \to 1^-}f(x) = \infty$ (d) As $~~x \to 1^+~~$, the value of $\frac{2}{x}$ approaches $2$ while $\frac{1}{ln~x}$ approaches $\infty$ $\lim\limits_{x \to 1^-}f(x) = -\infty$ (e) We can see a sketch of the graph below.
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