# Thread: Prove a limit using the delta epsilon method

1. ## Prove a limit using the delta epsilon method

Hello,
How can I prove that the limit of the following function equals to zero, using the delta epsilon method?
$\displaystyle \lim_{x\rightarrow 2}(x^2-4)=0$

I have reached this step, but I do not know what to do next:
$\displaystyle |x-2|< \frac{\varepsilon}{|x+2|}$

2. Originally Posted by msokol89
Hello,
How can I prove that the limit of the following function equals to zero, using the delta epsilon method?
$\displaystyle \lim_{x\rightarrow 2}(x^2-4)=0$
I have reached this step, but I do not know what to do next:
$\displaystyle |x-2|< \frac{\varepsilon}{|x+2|}$
If you know that $\displaystyle |x-2|<1$ then we know that $\displaystyle |x+2|<5$.
Pick $\displaystyle \delta = \min \left\{ {1,\frac{\varepsilon }{5}} \right\}$.

3. Hint :

In $\displaystyle [1,3]$ we have $\displaystyle |x+2|\leq 5$ so, choose $\displaystyle \delta=\min \{1,\epsilon/5\}$

Edited: Sorry, I didn't see Plato's post.

4. I understand, thanks to you both