Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 3 - Applications of Differentiation - 3.4 Limits at Infinity; Horizontal Asymptotes - 3.4 Exercises - Page 242: 23



Work Step by Step

Given $$ \lim _{x \rightarrow \infty}(\sqrt{x^{2}+a x}-\sqrt{x^{2}+b x}) $$ Then \begin{aligned} \lim _{x \rightarrow \infty}(\sqrt{x^{2}+a x}-\sqrt{x^{2}+b x}) &=\lim _{x \rightarrow \infty} \frac{(\sqrt{x^{2}+a x}-\sqrt{x^{2}+b x})(\sqrt{x^{2}+a x}+\sqrt{x^{2}+b x})}{\sqrt{x^{2}+a x}+\sqrt{x^{2}+b x}} \\ &=\lim _{x \rightarrow \infty} \frac{\left(x^{2}+a x\right)-\left(x^{2}+b x\right)}{\sqrt{x^{2}+a x}+\sqrt{x^{2}+b x}}\\ &=\lim _{x \rightarrow \infty} \frac{[(a-b) x] / x}{(\sqrt{x^{2}+ax}+\sqrt{x^{2}+b x}) / \sqrt{x^{2}}} \\ &=\lim _{x \rightarrow \infty} \frac{a-b}{\sqrt{1+a / x}+\sqrt{1+b / x}}\\ &=\frac{a-b}{\sqrt{1+0}+\sqrt{1+0}}\\ &=\frac{a-b}{2} \end{aligned}
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