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: 34



Work Step by Step

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