I would like to know if it is possible generalize this result and how it could be shown:
we know that
$ \begin{align} \lim_{x\to 0^+}x^x & =1;\\ \\ \lim_{x\to 0^+}x^{x^x} & =0;\\ \\ \lim_{x\to 0^+}x^{x^{x^x}}& =1 \end{align} $
$\begin{matrix}\displaystyle\lim_{x\to 0^+} \overbrace{x^{x^{x^{x^{\cdots^{x}}}}}}^{n\text{ times}} \end{matrix}\quad$
I would like to conclude that if $n$ equal limit is $1,$ and if $n$ is odd limit is $0$ is it Possible?
tanks in advances