Answer
See explanations.
Work Step by Step
Step 1. For a given small value $\epsilon\gt0$, we need to find a value $\delta\gt0$ so that for all $x$ in the interval of $2\lt x\lt2+\delta$, we get $|\frac{x-2}{|x-2|}-1|\lt\epsilon$
Step 2. As $x\gt2$, we have $|x-2|=x-2$; the last inequality becomes $|\frac{x-2}{x-2}-1|\lt\epsilon$ which gives $0\lt\epsilon$
Step 3. As the result from step 2 is always true, we can set $\delta$ to be any positive value such as $\delta=\epsilon$ so that for all $x$ in the interval of $2\lt x\lt2+\delta$, we get $|\frac{x-2}{|x-2|}-1|\lt\epsilon$, which proves the limit statement.
