Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 3 - Section 3.3 - Derivatives of Trigonometric Functions - 3.3 Exercises: 45

Answer

$$\lim\limits_{\theta\to0}\frac{\sin\theta}{\theta+\tan\theta}=\frac{1}{2}$$

Work Step by Step

$$A=\lim\limits_{\theta\to0}\frac{\sin\theta}{\theta+\tan\theta}$$ $$A=\lim\limits_{\theta\to0}\frac{\sin\theta}{\theta+\frac{\sin\theta}{\cos\theta}}$$ Divide both numerator and denominator by $\theta$ $$A=\lim\limits_{\theta\to0}\frac{\frac{\sin\theta}{\theta}}{1+\frac{\sin\theta}{\theta\cos\theta}}$$ $$A=\frac{\lim\limits_{\theta\to0}\frac{\sin\theta}{\theta}}{{\lim\limits_{\theta\to0}}(1+\frac{\sin\theta}{\theta\cos\theta})}$$ $$A=\frac{1}{1+\lim\limits_{\theta\to0}(\frac{\sin\theta}{\theta}\times\frac{1}{\cos\theta})}$$ $$A=\frac{1}{1+(1\times\frac{1}{\cos0})}$$ $$A=\frac{1}{1+(1\times1)}$$ $$A=\frac{1}{2}$$
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