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 98: 84



Work Step by Step

\begin{aligned} \lim _{x \rightarrow \infty}(\sqrt{9 x^{2}-x}-3 x) &=\lim _{x \rightarrow \infty}[\sqrt{9 x^{2}-x}-3 x] \cdot\left[\frac{\sqrt{9 x^{2}-x}+3 x}{\sqrt{9 x^{2}-x}+3 x}\right]\\ &=\lim _{x \rightarrow \infty} \frac{\left(9 x^{2}-x\right)-\left(9 x^{2}\right)}{\sqrt{9 x^{2}-x+3 x}}\\ &=\lim _{x \rightarrow \infty} \frac{-x}{\sqrt{9 x^{2}-x}+3 x} \\ &=\lim _{x \rightarrow \infty} \frac{\frac{-x}{x}}{\sqrt{\frac{9 x^{2}-x}{x^{2}}+\frac{3 x}{x}}}\\ &=\lim _{x \rightarrow \infty} \frac{-1}{\sqrt{9-\frac{1}{x}}+3}\\ &=\frac{-1}{3+3}\\ &=-\frac{1}{6} \end{aligned}
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