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: 36



Work Step by Step

\begin{align*} \lim _{x \rightarrow-\infty} \frac{4-3 x^{3}}{\sqrt{x^{6}+9}}&=\lim _{x \rightarrow-\infty} \frac{\left(4-3 x^{3}\right) / \sqrt{x^{6}}}{\sqrt{x^{6}+9} / \sqrt{x^{6}}}\\ &=\lim _{x \rightarrow-\infty} \frac{\left(4-3 x^{3}\right) /\left(-x^{3}\right)}{\sqrt{\left(x^{6}+9\right) / x^{6}}}\\ &=\lim _{x \rightarrow \infty} \frac{\left(-4 / x^{3}+3\right)}{\sqrt{1+9 / x^{6}}}\\ &=\frac{(0+3)}{\sqrt{1+0}}\\ &=3 \end{align*}
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