$\lim\limits_{x \rightarrow 2} \frac{x-2}{|x-2|}$ equals

none of these

$\lim\limits_{x \rightarrow 2^{+}} \frac{x-2}{|x-2|}=\lim\limits_{h \rightarrow 0} \frac{2+h-2}{|2+h-2|}=\frac{h}{|h|}=\frac{h}{h}=1$

$\lim\limits_{x \rightarrow 2^{-}} \frac{x-2}{|x-2|}=\lim\limits_{h \rightarrow 0} \frac{2-h-2}{|2-h-2|}=\frac{-h}{|-h|}=\frac{-h}{h}=-1$

LHL ≠ RHL

Limit does not exist.

Hence (D) is the correct answer.