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: 43



Work Step by Step

\begin{align*} \lim _{x\rightarrow 0} \frac{ \sin 3x\sin 2x}{x\sin 5x}&=\lim _{x\rightarrow 0} \frac{6}{5} \frac{ \sin 3x }{3x} \frac{ \sin 2x}{2x } \frac{ 5x}{\sin 5x}\\ &= \frac{6}{5} \lim _{3x\rightarrow 0} \frac{ \sin 3x }{3x} \lim _{2x\rightarrow 0} \frac{ \sin 2x}{2x } \lim _{5x\rightarrow 0} \frac{ 5x}{\sin 5x} \\ &= \frac{6}{5}. \end{align*} Where we used Theorem 2 -- that is, $\lim _{x\rightarrow 0}\frac{ \sin x}{ x}=1. $
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