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: 30



Work Step by Step

We have $$ \lim _{x \rightarrow \frac{\pi}{2}} \left(x- \frac{\pi}{2}\right) \tan x=0\times\infty $$ is an intermediate form, then we can apply L’Hôpital’s Rule as follows $$\lim _{x \rightarrow \frac{\pi}{2}} \left(x- \frac{\pi}{2}\right) \tan x=\lim _{x \rightarrow \frac{\pi}{2}} \frac{x- \frac{\pi}{2}}{\cot x}=\lim _{x \rightarrow \frac{\pi}{2}} \frac{1}{-\csc^2 x}=-1.$$
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