# Chapter 3 - Section 3.3 - Derivatives of Trigonometric Functions - 3.3 Exercises - Page 197: 42

The limit is 0.

#### Work Step by Step

Limit (L) $= \lim\limits_{\theta \to 0} \frac{\cos{\theta}-1}{\sin{\theta}}$ $= \lim\limits_{\theta \to 0} \frac{\cos{\theta}-1}{\theta} \times \frac{\theta}{\sin{\theta}}$ $= \lim\limits_{\theta \to 0} \frac{\cos{\theta}-1}{\theta} \times \lim\limits_{\theta \to 0} \frac{\theta}{\sin{\theta}}$ By equation 3: $$\lim\limits_{\theta \to 0} \frac{\cos{\theta}-1}{\theta}=0$$ And by equation 2: $$\lim\limits_{\theta \to 0} \frac{\sin{\theta}}{\theta}=1$$ Thus, L $=0\times1^{-1}=0\times1=0$ Therefore, the limit is 0.

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