# Thread: epsilon delta continuity applied to polynomial

1. ## epsilon delta continuity applied to polynomial

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!

2. We know that $\left| {x - 3} \right| < 1\, \Rightarrow \,\left| x \right| < 4\;\& \,\left| {x + 1} \right| < 5$.
So in picking $\delta$ make sure it is less than one so the above will true.
Given $\varepsilon>0$ pick $\delta = \min \left\{ {1,\frac{\varepsilon }
{{20}}} \right\}$

3. Thanks, I didn't consider that