Hi, I've been asked to prove that f(x) = x^3 - 9x +2 is continous at x=3 by using epsilon delta continuity. I assume I'm not allowed to use the theorem that the sum of continuous functions is continious.

So let e>0

Then |f(x)-f(3)| < e

|x^3 - 9x| < e

| x (x+3) (x-3) | < e

How do I find the requirements for delta so that |(x-3)| < delta?

I don't think I'm allowed to divide the inequality by x(x+3) as I understand that delta has to be independent of x.

It's clear that there is a solution for delta because cubic polynomials have solutioins, but it's very messy to solve.

What other way can I find delta?

Thanks!