## Calculus (3rd Edition)

Published by W. H. Freeman

# Chapter 2 - Limits - 2.6 Trigonometric Limits - Exercises - Page 77: 39

#### Answer

$$\frac{-3}{4}$$

#### Work Step by Step

\begin{align*} \lim _{\theta \rightarrow 0} \frac{ \sin (-3\theta)}{\sin (4\theta)}&=\lim _{\theta \rightarrow 0} \frac{ \sin (-3\theta)}{\sin 4\theta}\\ &=\lim _{\theta \rightarrow 0} \frac{-3}{4}\frac{ \sin (-3\theta)}{-3\theta}\frac{ 4\theta}{\sin 4\theta}\\ &= \frac{-3}{4}\lim _{-3\theta \rightarrow 0}\frac{ \sin (-3\theta)}{-3\theta}\lim _{4\theta \rightarrow 0}\frac{ 4\theta}{\sin 4\theta}\\ &= \frac{-3}{4}. \end{align*} Where we used Theorem 2 -- that is, $\lim _{x\rightarrow 0}\frac{ \sin x}{ x}=1.$

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