Answer
(a) $x \gt 100$
(b) $\lim\limits_{x \to \infty}\frac{1}{x^2} = 0$
Work Step by Step
(a) $\frac{1}{x^2} \lt 0.0001$
$x^2 \gt \frac{1}{0.0001}$
$x^2 \gt 10,000$
$x \gt 100$
(b) Let $f(x) =\frac{1}{x^2}$
This function is defined on the interval $(0, \infty)$
Let $\epsilon \gt 0$ be given.
Let $N = \sqrt{\frac{1}{\epsilon}}$
Suppose that $x \gt N$
Then:
$\vert \frac{1}{x^2} - 0\vert \lt \vert \frac{1}{N^2}\vert = \frac{1}{(\sqrt{\frac{1}{\epsilon}})^2} = \epsilon$
Therefore, $\lim\limits_{x \to \infty}\frac{1}{x^2} = 0$
