sequences and series - How to solve this indetermination when calculating a limit?

I'm stuck trying to find the limit of the sequence

$$\frac{\sqrt{12 + a_n} - \sqrt{4a_n}}{a_n^2 - 2a_n - 8}$$

Where I'm given that $a_n > 4$ and $a_n \rightarrow 4$

Both the numerator and the denominator tend to 0, and I can't find how to solve this indetermination. I tried multiplying and dividing by the "reciprocal" of the numerator to get rid of the square roots in the numerator, but that doesn't seem to lead anywhere. What else can I try?

Answer

Hint:

$$b^2-2b-8=(b-4)(b+2)$$

$$\sqrt{12+b}-\sqrt{4b}=-\dfrac{3(b-4)}{\sqrt{12+b}+\sqrt{4b}}$$

If $b\to4,b\ne4\implies b-4\ne0$ hence can be cancelled safely