University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 2 - Section 2.6 - Limits Involving Infinity; Asymptotes of Graphs - Exercises - Page 107: 5

Answer

$\lim_{x\to\infty}f(x)=\lim_{x\to-\infty}f(x)=1/2$
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Work Step by Step

$$g(x)=\frac{1}{2+\frac{1}{x}}$$ (a) As $x\to\infty$, $1/x$ will approach $0$. Therefore, $$\lim_{x\to\infty}f(x)=\lim_{x\to\infty}\frac{1}{2+\frac{1}{x}}=\frac{1}{2+0}=\frac{1}{2}$$ (b) As $x\to-\infty$, $1/x$ will approach $-1/\infty$ and this approaches $0$ as well. Therefore, $$\lim_{x\to-\infty}f(x)=\lim_{x\to-\infty}\frac{1}{2+\frac{1}{x}}=\frac{1}{2+0}=\frac{1}{2}$$ A graph of the function $f(x)$ is enclosed below, which shows that $f(x)$ approaches $1/2$ as $x$ approaches either $\infty$ or $-\infty$.
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