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: 82



Work Step by Step

\begin{aligned} \lim _{x \rightarrow-\infty}(\sqrt{x^{2}+3}+x) &=\lim _{x \rightarrow-\infty}[\sqrt{x^{2}+3}+x] \cdot\left[\frac{\sqrt{x^{2}+3}-x}{\sqrt{x^{2}+3}-x}\right]\\ &=\lim _{x \rightarrow-\infty} \frac{\left(x^{2}+3\right)-\left(x^{2}\right)}{\sqrt{x^{2}+3}-x}\\ &=\lim _{x \rightarrow-\infty} \frac{3}{\sqrt{x^{2}+3-x}} \\ &=\lim _{x \rightarrow-\infty} \frac{\frac{3}{\sqrt{x}}}{\sqrt{1+\frac{3}{x^{2}}} \frac{x}{\sqrt{x^{2}}}}\\ &=\lim _{x \rightarrow-\infty} \frac{-\frac{3}{x}}{\sqrt{1+\frac{3}{x^{2}}}+1}\\ &=\frac{0}{1+1}\\ &=0 \end{aligned}
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