Consider the following two functions involving power towers; $$f(x)=\arcsin(x)\uparrow\uparrow(2k)$$ $$g(x)=\arcsin(x)\uparrow\uparrow(2k-1)$$ Where $k\in \mathbb{Z}^+$. The global minimum of $f(x)$ can be described as being point $P(a,b)$. What would happen to the values of $a$ and $b$ as $k \to \infty$? Will they both approach $0$ or will they converge to a non-zero number?
(Note: I've included function $g(x)$ to act as a comparison to $f(x)$; I don't believe it's actual required to find the solution for this question)
The $\arcsin$ doesn't really do anything. Consider instead:
$$f_n(x)=x\uparrow\uparrow n$$
We have for even $n$,
$$\lim_{n\to\infty}f_{2n}(x)=\begin{cases}f_e(x),&0<x<e^{-e}\\f(x),&e^{-e}\le x\le e^{1/e}\end{cases}$$
and for odd $n$,
$$\lim_{n\to\infty}f_{2n+1}(x)=\begin{cases}f_o(x),&0<x<e^{-e}\\f(x),&e^{-e}\le x\le e^{1/e}\end{cases}$$
where $f^{-1}(x)=\sqrt[x]x$ and $f_e$ and $f_o$ are two different branches to the solution of $y=x^{x^y}$ which are not solutions to $y=x^y$.
Graph of the limit.
So the limit of the infinums are $0$ for odds and $f(e^{-e})=e^{-1}$ for the evens.