Answer
See explanations.
Work Step by Step
Step 1. To prove $\lim_{x\to 2^-}\frac{1}{x-2}=-\infty$ , for every negative real number $-B$, we need to find a corresponding number $\delta\gt0$ such that for all x, $-\delta\lt x-2 \lt 0$, we get $\frac{1}{x-2}\lt -B$
Step 2. The last inequality gives $x-2 \gt -\frac{1}{B}$, thus we can choose $\delta=\frac{1}{B}$ so that when we go back in the steps, we see that for all x, $-\delta\lt x-2 \lt 0$, we get $\frac{1}{x-2}\lt -B$ which proves the limit statement.
