Application of the Limit Theorem using Epsilon-delta

Question

I am trying to apply the limit definition to $f(x) = \frac{1}{x^2}$ over a set, $S = [a,b] $ given that $a > 0 $. I need to find the $\delta_\epsilon$ in terms of a and b. Would anybody mind helping me out here?

Answer 1

Let $x_0\in [a,b]$. Then, we have for $x\ne x_0$

$$0<\left|\frac{1}{x^2}-\frac{1}{x_0^2}\right|=\left|\frac{x^2-x_0^2}{x^2x_0^2}\right|\le\frac{2b}{a^4}|x-x_0|<\epsilon$$

whenever $0<|x-x_0|<\delta=\frac{a^4}{2b}\epsilon$. Therefore, $1/x^2$ is uniformly continuous for $0<a\le x\le b$.