Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 2: Limits and Continuity - Section 2.6 - Limits Involving Infinity; Asymptotes of Graphs - Exercises 2.6 - Page 97: 22

Answer

(a) $ -\infty $ (b) $\infty $

Work Step by Step

(a) \begin{align*} \lim_{x\to \infty}f(x)&= \lim_{x\to \infty} \frac{5x^8-2x^3+9}{ 3+x-4x^5}\\ &=\lim_{x\to \infty} \frac{5x^8/x^5-2x^3/x^5+9/x^5}{ 3/x^5+x/x^5-4x^5/x^5}\\ &= \frac{5\lim_{x\to \infty}( x^3)-2\lim_{x\to \infty}(1/x^2)+9\lim_{x\to \infty}(1/x^5)}{\lim_{x\to \infty}( 3/x^5)+\lim_{x\to \infty}(1/x^4)-4\lim_{x\to \infty}(1)}\\ &= \frac{5\lim_{x\to \infty}( x^3) }{-4 }\\ &= -\infty \end{align*} (b) \begin{align*} \lim_{x\to -\infty}f(x)&= \lim_{x\to- \infty} \frac{5x^8-2x^3+9}{ 3+x-4x^5}\\ &=\lim_{x\to -\infty} \frac{5x^8/x^5-2x^3/x^5+9/x^5}{ 3/x^5+x/x^5-4x^5/x^5}\\ &= \frac{5\lim_{x\to -\infty}( x^3)-2\lim_{x\to -\infty}(1/x^2)+9\lim_{x\to -\infty}(1/x^5)}{\lim_{x\to- \infty}( 3/x^5)+\lim_{x\to- \infty}(1/x^4)-4\lim_{x\to -\infty}(1)}\\ &= \frac{5\lim_{x\to \infty}( x^3) }{-4 }\\ &= \infty \end{align*}
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