Answer
$$\frac{1}{4}$$
Work Step by Step
We have
$$
\lim _{x \rightarrow 4} \left(\frac{1}{\sqrt{x}-2}-\frac{4}{x-4}\right)=\infty-\infty.
$$
is an intermediate form, then we can apply L’Hôpital’s Rule as follows
$$
\lim _{x \rightarrow 4} \left(\frac{1}{\sqrt{x}-2}-\frac{4}{x-4}\right)=\lim _{x \rightarrow 4} \frac{\sqrt{x}+2-4}{x-4}=\lim _{x \rightarrow 4} \frac{\sqrt{x}-2}{x-4}=\lim _{x \rightarrow 4} \frac{1/(2\sqrt{x})}{1}=\frac{1}{4}.
$$
