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.