# Evaluate the following limits. lim_{thetarightarrow0}frac{cos^2theta-1}{theta}

Question
Limits and continuity
Evaluate the following limits.
$$\lim_{\theta\rightarrow0}\frac{\cos^2\theta-1}{\theta}$$

2020-11-15
Given that $$\lim_{\theta\rightarrow0}\frac{\cos^2\theta-1}{\theta}$$
Compute the limit as follows.
$$\lim_{\theta\rightarrow0}\frac{\cos^2\theta-1}{\theta}=\lim_{\theta\rightarrow0}\frac{-(1-\cos^2\theta)}{\theta}$$
$$=\lim_{\theta\rightarrow0}\frac{-\sin^2\theta}{\theta}$$
$$=\lim_{\theta\rightarrow0}\frac{\sin\theta}{\theta}\times\lim_{\theta\rightarrow0}(-\sin\theta)$$
$$=1\times(0)$$
$$=0$$
Thus, $$\lim_{\theta\rightarrow0}\frac{\cos^2\theta-1}{\theta}=0$$

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