# Interpreting a limit as a derivative

I am being stumped by the following question:

Evaluate the limit by interpreting each as a derivative:

$$\lim_{x\,\to\tfrac{\pi}{6}} \frac{\cos(2x) - \frac{1}{2}}{x - \frac{\pi}{6}}$$

The only way I can think to solve this is using L'Hopital's rule. I have done that and got the correct answer $-\sqrt{3}$. But, I can not figure out how to do it the was it is described.

Any help would be greatly appreciated.

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Hint: Write down the definition of the derivative of $f(x)=\cos 2x$ at $\frac{\pi}6$