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## Homework Statement

"Use the given graph of f(x) = x^2 to find a number delta such that..."

## Homework Equations

If ##([\lim_{x \rightarrow a} f(x)] - L)## then there exists an ##\epsilon## such that ##[0 < |x - a| < \delta] \Rightarrow [|f(x) - L| < \epsilon]##.

Here, f(x) and ##\epsilon## are given. So we have to find a ##\delta## such that ##|x-1|<\delta## implies ##|x^{2}-1| < \frac{1}{2}##.

## The Attempt at a Solution

I assume that we have to relate ##x^{2}-1## with ##\delta## somehow. I tried to do it like this:

Prove that ##(|x-1| < \delta) \Rightarrow (|x^{2}-1|<\frac{1}{2})##.

Choose ##\delta##:

##|x^{2}-1|<\frac{1}{2}##

##\Rightarrow -\frac{1}{2}<x^{2}-1<\frac{1}{2}##

##\Rightarrow -\frac{1}{2}+1<x^{2}<\frac{1}{2}+1##

##\Rightarrow \frac{1}{2}<x^{2}<\frac{3}{2}##

##\Rightarrow \sqrt{\frac{1}{2}}<x<\sqrt{\frac{3}{2}}##

But here I get stuck. I feel that I have to relate this to ##\delta##, but I don't know how!

As an aside, I can do regular ##\delta-\epsilon## proofs with linear equations now, so by rights this should be easy. But for some reason, it isn't.