My limit challenge problem

[lookagain]

I made this problem.

Calculate this:

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



\(\displaystyle You \ may \ use \ \lim_{x \to 0^+}(x^x) \ = \ 1.\)

 

[tkhunny]

Seems to be screaming for the binomial theorem.

 

[lookagain]

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

\(\displaystyle Let \ y \ belong \ to \ the \ positive \ Reals.\)

\(\displaystyle You \ may \ use \ \lim_{y \to 0^+}(y^y) \ = \ 1. ** \ \ \ \ \ Edit \ on \ the \ variable\)

Hints:
-------

For 0 < x < 1, x^x < 1 . . . . . . Why?

As x approaches 0 from the right, (1 - x^x) approaches 0 from above . . . . . Why?


\(\displaystyle Then \ \lim_{x \to 0} (1 - x^x)^x \ is \ of \ the \ form \ of **\)