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: 24



Work Step by Step

\begin{align*} \lim _{x \rightarrow-\infty}\left(\frac{x^{2}+x-1}{8 x^{2}-3}\right)^{1 / 3}&=\lim _{x \rightarrow-\infty}\left(\frac{x^{2}/x^{2}+x/x^{2}-1/x^{2}}{8 x^{2}/x^{2}-3/x^{2}}\right)^{1 / 3}\\ &=\lim _{x \rightarrow-\infty}\left(\frac{1+1/x -1/x^{2}}{8 -3/x^{2}}\right)^{1 / 3}\\ &=\left(\frac{\lim _{x \rightarrow-\infty}(1)+\lim _{x \rightarrow-\infty}(1/x) -\lim _{x \rightarrow-\infty}(1/x^{2})}{\lim _{x \rightarrow-\infty}(8 ) -\lim _{x \rightarrow-\infty}(3/x^{2})}\right)^{1 / 3}\\ &=\left(\frac{1+0-0}{8-0}\right)^{1 / 3}\\ &=\frac{1}{2} \end{align*}
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