Answer
The limit does not exist.
Work Step by Step
The one-sided limits,
$$\lim _{h \rightarrow 0^+}\frac{\sqrt{2+h}-2}{h}\frac{\sqrt{2+h}+2}{\sqrt{2+h}+2}=\lim _{h \rightarrow 0^+}\frac{h-2}{h\sqrt{2+h}+2}=\infty $$
$$\lim _{h \rightarrow 0^-}\frac{\sqrt{2+h}-2}{h}\frac{\sqrt{2+h}+2}{\sqrt{2+h}+2}=\lim _{h \rightarrow 0^-}\frac{h-2}{h\sqrt{2+h}+2}=-\infty $$
Hence the one-sided limits are not equal and infinite. The overall limit does not exist.
