While messing around on Desmos calculator, I found an interesting factorial/tetration function, where we are using Knuth arrow notation.
Where there are $n$ factorials following $x$ in $f(x,n)$.
Looking at it graphically, you may find the link above, it appears there are two roots, $x=1,a$, where $a$ seems to get increasingly closer to $2$ as $n$ goes to infinity.
However, I can't figure out how to prove this because limits like this are not the sort I can deal with.
I do notice, however, that the following is the infinite power tower:
And the inverse of the infinite power tower is:
So the real problem is probably in the my factorial function $f(x,n)$, which I find hard to deal with.
Any solutions or directions to point?
EDIT: It appears as though this root, $x=a$, will most likely occur higher than $2$, which is going to give us a problem. The infinite power tower doesn't hold for $x>\sqrt[e]e$, which means we might have to do this some harder way...