Calculus (3rd Edition)

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

Chapter 2 - Limits - 2.5 Evaluating Limits Algebraically - Exercises - Page 72: 32



Work Step by Step

\begin{aligned} \lim _{\theta \rightarrow \frac{\pi}{2}}(\sec \theta -\tan \theta )&=\lim _{\theta \rightarrow \frac{\pi}{2}} \frac{1-\sin \theta}{\cos \theta} \cdot \frac{1+\sin \theta}{1+\sin \theta}\\ &=\lim _{\theta \rightarrow \frac{\pi}{2}} \frac{1-\sin ^{2} \theta}{\cos \theta(1+\sin \theta)}\\ &=\lim _{\theta \rightarrow \frac{1}{2}} \frac{\cos ^{2} \theta}{\cos \theta(1+\sin \theta)}\\ &=\lim _{\theta \rightarrow \frac{\pi}{2}} \frac{\cos \theta}{1+\sin \theta}\\ &=\frac{0}{2}=0 \end{aligned}
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