Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 2 - Derivatives - Review - Exercises - Page 197: 46

Answer

$$\lim_{t\to0}\frac{t^3}{\tan^32t}$$

Work Step by Step

Given $$\lim_{t\to0}\frac{t^3}{\tan^32t}$$ Since $$ \lim_{t\to0}\frac{0^3}{\tan^32(0)}=\frac{0}{0}$$ then \begin{align*} \lim_{t\to0}\frac{t^3}{\tan^32t}&=\lim_{t\to0}\frac{1}{\frac{\tan^32t}{t^3}}\\ &=\frac{\lim_{t\to0}1}{\lim_{t\to0}\frac{\tan^32t}{t^3}}\\ &=\frac{\lim_{t\to0}1}{\left(2\lim_{2t\to0}\frac{\tan 2t}{2t }\right)^3}\\ &=\frac{1}{8} \end{align*}
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