Answer
$\displaystyle \frac{1}{\sqrt{x+h-2}+\sqrt{x-2}}$
Work Step by Step
$f(x)=\sqrt{x-2}$
$\displaystyle \frac{f(x+h)-f(x)}{h}=\frac{\sqrt{x+h-2}-\sqrt{x-2}}{h}$
... use the hint, rationalize with $\displaystyle \frac{\sqrt{x+h-2}+\sqrt{x-2}}{\sqrt{x+h-2}+\sqrt{x-2}}$
$=\displaystyle \frac{(x+h-2)-(x-2)}{h(\sqrt{x+h-2}+\sqrt{x-2})}$
$=\displaystyle \frac{h}{h(\sqrt{x+h-2}+\sqrt{x-2})}$ ... h cancels
$=\displaystyle \frac{1}{\sqrt{x+h-2}+\sqrt{x-2}}$
