Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 2 - Limits - 2.2 Limits: A Numerical and Graphical Approach - Exercises - Page 55: 55

Answer

$$2.5$$

Work Step by Step

We need to find $$\lim _{\theta \rightarrow 0} \frac{\sin 5 \theta}{\sin 2 \theta}$$ Consider the function $$ f(\theta)= \frac{\sin 5 \theta}{\sin 2 \theta}$$ From the following figure, we can observe that \begin{align*} \lim _{\theta \rightarrow 0^+} f( \theta)&=2.5\\ \lim _{\theta \rightarrow 0^-} f( \theta)&=2.5 \end{align*} Since the left and right limits equal each other, we can say that $$\lim _{\theta \rightarrow 0} \frac{\sin 5 \theta}{\sin 2 \theta}=2.5 $$
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