Limit -- My own problem

[lookagain]

Using the fact that as $\displaystyle \lim_{x\to0^+}(x^x) = 1$ from below for sufficiently
small positive real numbers $\displaystyle x$, determine the following limit:


$\displaystyle \lim_{x\to 0^+}[1 - (1 - x^x)^x]^x$

 

[galactus]

It would appear since $\displaystyle \lim_{x\to 0^{+}}x^{x}=1$ that

Try rewrting as:

$\displaystyle e^{\huge\lim_{x\to 0^{+}}xln\left(1-(1-x^{x})^{x}\right)}$

The limit in the power of e approaches 0 and we get $\displaystyle e^{0}=1$

The limit is 1 as well.