Solving indetermination in limit I'm trying to solve this limit, but I can't get it out of a $\frac{0}{0}$ indetermination:
$$\displaystyle \lim_{x \to 4} \; \frac{x-4}{5-\sqrt{x^2+9}}$$
Maybe there is something I'm missing. Thanks a lot in advanced.

Alternatively, if you know derivatives, notice that
$$\lim_{x\to 4}\frac{\sqrt{x^2+9}-5}{x-4}$$
is, by definition, $f'(4)$ with  $f(x) = \sqrt{x^2+9}$. Since 
$$f'(x) = \frac{x}{\sqrt{x^2+9}}$$
then $f'(4) = \frac{4}{5}$. The limit you want is the negative reciprocal, since
$$\frac{x-4}{5 - \sqrt{x^2+9}} = - \frac{1}{\quad\frac{\sqrt{x^2+9}-5}{x-4}\quad}.$$
