Prove that for defined as , has a limit at .

Obviously the limit is . So we need to show that for each , there exists such that for each satisfying:

it is then true that:

So let's get busy. The last bit simplifies to:

I can only guess what to do next. I could try this:

which simplifies to:

But this is a little strange. I want to take the square root, but the left side could be negative, so I can't. So, maybe I could do:

which simplifies to:

But how would I get that into something workable?

Alternatively, we could go back to this:

And try this:

But then we have a problem of negatives. So let's assume . Then:

Now assume . Then:

In both cases we have this:

But still we have a problem. How do we find from this:

?

Or am I going about this entirely the wrong way?

Any help would be much appreciated!