Answer
$-\frac{\sqrt{3}}{2}$
Work Step by Step
$\lim\limits_{\theta \to \pi/3}\frac{cos~\theta-0.5}{\theta-\pi/3}$
This is an expression for the derivative of $~~f(\theta) = cos~\theta~~$ evaluated at the point $\theta = \frac{\pi}{3}$
We can evaluate the derivative:
$f(\theta) = cos~\theta$
$f'(\theta) = -sin~\theta$
$f'(\frac{\pi}{3}) = -sin~\frac{\pi}{3}$
$f'(\frac{\pi}{3}) = -\frac{\sqrt{3}}{2}$