Calculus (3rd Edition)

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

Chapter 2 - Limits - Chapter Review Exercises - Page 95: 50



Work Step by Step

We evaluate the limit: \begin{aligned} \lim_{\theta \to 0}\frac{\tan \theta-\sin \theta}{\sin ^{3} \theta} &=\lim_{\theta \to 0}\frac{\frac{\sin \theta}{\cos \theta}-\sin \theta}{\sin ^{3} \theta} \\ &=\lim_{\theta \to 0}\frac{\frac{\sin \theta-\sin \theta \cos \theta}{\cos \theta}}{\sin ^{3} \theta} \\ &=\lim_{\theta \to 0}\frac{\sin \theta-\sin \theta \cos \theta}{\sin ^{3} \theta \cdot \cos \theta}\\ &=\lim_{\theta \to 0}\frac{1- \cos \theta}{\sin ^{2} \theta \cdot \cos \theta}\\ &=\lim_{\theta \to 0}\frac{1- \cos \theta}{(1-\cos ^{2} \theta) \cdot \cos \theta}\\ &=\lim_{\theta \to 0}\frac{1 }{(1+\cos \theta) \cdot \cos \theta}\\ &=\frac{1}{2} \end{aligned}
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