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: 27


(a) $L=\lim _{\theta \rightarrow 0} \frac{14 \sin \theta}{\theta}$ (b) $L=14$.

Work Step by Step

Given $$ L= \lim _{x \rightarrow 0} \frac{\sin 14x}{x}. $$ (a) Putting $\theta =14x $, when $ x\to 0$ then $\theta \to 0$ . Hence, we have $$ L= \lim _{x \rightarrow 0} \frac{\sin 14x}{x}=\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta/14}=\lim _{\theta \rightarrow 0} \frac{14 \sin \theta}{\theta}. $$ (b) $$ L= \lim _{\theta \rightarrow 0} \frac{14 \sin \theta}{\theta}=14 \lim _{\theta \rightarrow 0} \frac{ \sin \theta}{\theta}=14. $$ Where we used the fact that $\lim _{\theta \rightarrow 0} \frac{ \sin \theta}{\theta}=1$.
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