Answer
See explanations.
Work Step by Step
Step 1. To prove the limit $\lim_{x\to0}\frac{-1}{x^2}=-\infty$, for any large negative number $-B$, we need to find a corresponding value $\delta\gt0$ so that for all x in the interval $0\lt|x|\lt\delta$, we get $\frac{-1}{x^2}\lt -B$
Step 2. The last inequality gives $\frac{1}{x^2}\gt B$ and $x^2\lt 1/B$ which leads to $|x|\lt1/ \sqrt B$, thus we can choose $\delta=1/\sqrt B$ and when we go back in the steps, for all x in the interval $0\lt|x|\lt\delta$, we get $\frac{-1}{x^2}\lt -B$ which proves the limit statement.