$x^y=y^x$: prove that y approaches $1$ as x approaches $\infty$

Question

$y^x = x^y$

I graphed this function. I have two question about it:

1. Its graph consists of two curves, one of which is $x=y$ and the other curve looks like a hyperbola(in the first quadrant) and the curves meet at $(e,e)$. Why $(e,e)$?
2. $y$ value seems to approach $1$ as $x$ approaches $\infty$, how to prove that?

Answer 1

Parametric form for the "almost hyperbola" part is $$ x = v^{1/(v-1)} \\ y = v^{v/(v-1)} \tag1$$ with $0<v<+\infty$. (See HERE) Then take $v \to 0$ and $v \to +\infty$ to find the behavior in the two directions.

Also, to explain $(e,e)$, try to find $v$ in $(1)$ where $x=y$.