# How do you find the limit of (sqrt(x+4) -2) / x as x approaches 0?

$\frac{1}{4}$
We have limit of indeterminate form, ie $\frac{0}{0}$ so can use L'Hopital's rule:
${\lim}_{x \rightarrow 0} \frac{\sqrt{x + 4} - 2}{x} = {\lim}_{x \rightarrow 0} \frac{\frac{d}{\mathrm{dx}} \left(\sqrt{x + 4} - 2\right)}{\frac{d}{\mathrm{dx}} \left(x\right)}$
$= {\lim}_{x \rightarrow 0} \frac{\frac{1}{2 \sqrt{x + 4}}}{1} = \frac{1}{2 \sqrt{0 + 4}} = \frac{1}{4}$