Check correct delta in eps-delta proof

Question

I been stuck now with this seemingly simple exercise for some time.

I need to show that:

$|x^2-4| < \epsilon$ when $0 < |x-2| < \epsilon(5+\epsilon)^{-1}$

But I'm at a loss.

I know that I somehow have to recognize that $|x^2 - 4| = |x-2||x+2| < \epsilon$ and then use this with the other inequality (for the delta) to prove these inequalities hold.

I would strongly appreciate any help.

Thanks in advance!

Answer 1

Hint: If $|x-2| < { \epsilon \over 5+ \epsilon}$ then $|x+2| \le |x-2| + 4 \le { \epsilon \over 5+ \epsilon} + 4 = 5 ({ \epsilon+4\over \epsilon +5}) <5$.