EXAMPLE $5$ Use this definition to prove the following limit$.$

$\underset{x0}{lim}\frac{1}{x^2}=$

SOLUTION Let $M$ be a given positive number. We want to find a number $$ such that if then $\frac{1}{x^2}>M.$

But

So if we choose $=$ and then $\frac{1}{x^2}>M.$ This shows that $\frac{1}{x^2}$ as $x0.$