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!