Express $x^{x^{x^{x^{x...}}}}$ in the form of a limit.

Question

I want to rewrite $x^{x^{x^{x^{x...}}}}$ as a limit in the form of $$\lim_{n\to \infty}f(n)$$ I have defined the recurrence relation $$a_x(n) = x^{a_x(n-1)}$$ and $$a_x(1) = x$$ So, I can write the limit as: $$\lim_{n \to \infty}a_x(n)$$ but since my main goal is to evaluate the limit for $x = \sqrt2$, i.e $\lim_{n \to \infty}a_{\sqrt2}(n)$, this notation won't help. Please add some hints as to how should I proceed with solving the limit too. The page where I found this says that one may set $A = x^{x^{x^{x^{x...}}}}$, then $x^A = A$, so $x = A^{1\over A}$, which implies that for $A = 2,4$, $x = \sqrt2$. I have trouble finding which one is correct (using a calculator, I found it is very close to 2 for $n = 10$), but the page says something about finding the limit and leaves it there for the reader. Any help would be appreciated.