Substitution is perfectly appropriate, since the denominator does not evaluate to $0$. But I think you substituted $2 = f(x)$ instead of using $f(2) = 4$, and likewise, you substituted $g(x) =2$ instead of $g(2) = 9$.
$$\lim_{x\to 2} \frac{\sqrt{f(x) - 2}}{\sqrt{g(x) - 2}} = \frac {\sqrt{4 - 2}}{\sqrt{9-2}} = \sqrt{\frac{2}7}$$
ADDED: Even with the correction in formatting, substitution is perfectly appropriate to use, again, since the denominator does not evaluate to /approach $0$:
$$\lim_{x\to 2} \frac{\sqrt{f(x)} - 2}{\sqrt{g(x)} - 2} = \frac {\sqrt{4} - 2}{\sqrt{9} -2} = \frac{0}{1} = 0$$
;-)
– egreg May 10 at 20:21