Answer
$-\frac{\sqrt 3}{2}$
Work Step by Step
$$\lim\limits_{\theta \to \frac{\pi}{3}}\frac{\cos(\theta)-0.5}{\theta-\frac{\pi}{3}}$$
$$\lim\limits_{\theta \to \frac{\pi}{3}}\frac{\cos(\theta)-\frac{1}{2}}{\theta-\frac{\pi}{3}}$$
$$\lim\limits_{\theta \to \frac{\pi}{3}}\frac{\cos(\theta)-\cos(\frac{\pi}{3})}{\theta-\frac{\pi}{3}}=(\cos(x))'|_{x=\frac{\pi}{3}}=-\sin(\frac{\pi}{3})=-\frac{\sqrt 3}{2}$$
