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: 39



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. $
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.