Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 7 - Exponential Functions - 7.7 L'Hôpital's Rule - Exercises - Page 366: 18



Work Step by Step

We have $$ \lim _{x \rightarrow 4} \left(\frac{1}{\sqrt{x}-2}-\frac{4}{x-4}\right)=\infty-\infty. $$ is an intermediate form, then we can apply L’Hôpital’s Rule as follows $$ \lim _{x \rightarrow 4} \left(\frac{1}{\sqrt{x}-2}-\frac{4}{x-4}\right)=\lim _{x \rightarrow 4} \frac{\sqrt{x}+2-4}{x-4}=\lim _{x \rightarrow 4} \frac{\sqrt{x}-2}{x-4}=\lim _{x \rightarrow 4} \frac{1/(2\sqrt{x})}{1}=\frac{1}{4}. $$
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