Calculus (3rd Edition)

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

Chapter 2 - Limits - 2.6 Trigonometric Limits - Exercises - Page 77: 51


See the proof below.

Work Step by Step

\begin{align*} \lim _{\theta \rightarrow 0} \csc \theta -\cot\theta &= \lim _{\theta \rightarrow 0} \frac{1}{\sin \theta}-\frac{\cos \theta}{\sin \theta}\\ &=\lim _{\theta \rightarrow 0} \frac{1-\cos \theta}{\sin \theta}\\ &= \lim _{\theta \rightarrow 0} \frac{1-\cos \theta}{\theta} \frac{\theta}{\sin \theta}\\ &= \lim _{\theta \rightarrow 0} \frac{1-\cos \theta}{\theta} \lim _{\theta \rightarrow 0} \frac{\theta}{\sin \theta}\\ &=0. \end{align*} Where we used Theorem 2 -- that is, $\lim _{x\rightarrow 0}\frac{ \sin x}{ x}=1$ and $\lim _{x\rightarrow 0}\frac{ 1-\cos x}{x}=0. $
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