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 $-\delta\lt x\lt0$, we get $|\frac{x}{|x|}+1|\lt\epsilon$
Step 2. As $x\lt0$, we have $|x|=-x$; the last inequality becomes $|-\frac{x}{x}+1|\lt\epsilon$, which gives $0\lt\epsilon$
Step 3. As the result from step 2 is always true, we can set $\delta$ to be any value such as $\delta=\epsilon$ so that for all $x$ in the interval of $-\delta\lt x\lt0$, we get $|\frac{x}{|x|}+1|\lt\epsilon$, which proves the limit statement.
