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: 35



Work Step by Step

\begin{align*} \lim _{x \rightarrow \infty} \frac{x-3}{\sqrt{4 x^{2}+25}}&=\lim _{x \rightarrow \infty} \frac{(x-3) / \sqrt{x^{2}}}{\sqrt{4 x^{2}+25} / \sqrt{x^{2}}}\\ &=\lim _{x \rightarrow \infty} \frac{(x-3) / x}{\sqrt{\left(4 x^{2}+25\right) / x^{2}}}\\ &=\lim _{x \rightarrow \infty} \frac{(1-3 / x)}{\sqrt{4+25 / x^{2}}}\\ &=\frac{(1-0)}{\sqrt{4+0}}\\ &=\frac{1}{2} \end{align*}
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