What is the Limit of x^((x^x)-1) as x Approaches Zero?

[tsuwal]

Homework Statement

Limit of x^((x^x)-1) as x->0

Homework Equations

Lim x^x=1
x->0

The Attempt at a Solution

it's an 0^0 indetermination so I tried to solve it the usual way, by first calculating the limit of log(x)*(x^x-1) as x->0 with L'hopital's rule. I got e^0=1 confirmed by Wolfram. However, the using l'hospital's rule on this limit is not very practical, is there a better soluction?

 

[STEMucator]

Suppose you let y = x^((x^x)-1), what is ln(y) = ?

 

[tsuwal]

I would get the same limit log(x)*(x^x-1) as x->0 but solving this by L'hopital's rule takes a while or is it the fastest way?

 

[STEMucator]

Oh whoops, I read too quickly. Yeah I don't see how you would do this without LH otherwise.