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: 33



Work Step by Step

\begin{align*} \lim _{\theta \rightarrow 0} \frac{ \sin 7\theta}{\sin 3\theta}&=\lim _{\theta \rightarrow 0} \frac{ \sin 7\theta}{\sin 3\theta}\\ &=\lim _{\theta \rightarrow 0} \frac{7}{3}\frac{ \sin 7\theta}{7\theta}\frac{ 3\theta}{\sin 3\theta}\\ &= \frac{7}{3}\lim _{7\theta \rightarrow 0}\frac{ \sin 7\theta}{7\theta}\lim _{3\theta \rightarrow 0}\frac{ 3\theta}{\sin 3\theta}\\ &= \frac{7}{3} . \end{align*} Where we used Theorem 2 -- that is, $\lim _{x\rightarrow 0}\frac{ \sin x}{ x}=1. $
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