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: 28

Answer

$ \frac{9}{7} $

Work Step by Step

\begin{align*} \lim _{h\rightarrow 0} \frac{ \sin 9h}{\sin 7h}&=\lim _{h\rightarrow 0} \frac{9}{7} \frac{ \sin 9h}{ 9h} \frac{ 7h}{\sin 7h}\\ &= \frac{9}{7} \lim _{9h\rightarrow 0}\frac{ \sin 9h}{ 9h} \lim _{7h\rightarrow 0}\frac{ 7h}{\sin 7h}\\ &= \frac{9}{7} . \end{align*} Where we used Theorem 2 -- that is, $\lim _{x\rightarrow 0}\frac{ \sin x}{ x}=1. $
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