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: 54



Work Step by Step

\begin{align*} \lim _{h \rightarrow 0}\frac{\cos 3 h-1}{\cos 2h-1}&= \lim _{h \rightarrow 0} \frac{\cos 3 h-1}{h^2}\frac{h^2}{\cos 2h-1}\\ &= \lim _{h \rightarrow 0} \frac{\cos 3 h-1}{(3h)^2}\frac{(2h)^2}{\cos 2h-1}\\ &= \lim _{h \rightarrow 0} \frac{9}{4} \frac{1-\cos 3 h}{(3h)^2}\frac{(2h)^2}{1-\cos 2h}\\ &= \lim _{h \rightarrow 0}\frac{1-\cos 3 h}{(3h)^2}\frac{(2h)^2}{1-\cos 2h}\\ &= \frac{9}{4} \left(\frac{1}{2}\right)( 2).\\ &= \frac{9}{4}. \end{align*} Where we used the fact that $\lim _{h \rightarrow 0} \frac{1-\cos h}{h^2}=\frac{1}{2}$.
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