Proving Convergence of x^x to 1 at x->0

[prasannaworld]

I am trying to prove that the upper +limit of x^x, when x->0 converges to 1.

So I started by converting x^x to e^(x ln(x)). I know that this eliminates the domain: x <= 0, but I still believe that I can still continue on.

So here I tried to constrain the limit: x ln(x) (i.e. x->0, x ln(x) -> 0; which is where I failed. Although I can show via the L'hopital's Rule that it is true, I struggle to show it via the Epsilon Delta Definition.

I know that x^x is undefined at 0, but I still want to show that the curve converges towards 1.

 

[arildno]

Why not show that L'Hopitul's rule can be proven in general with the epsilon/delta-definition?