I just discovered these after approximation by Borwein and Bailey for $\pi$
And I'm assuming the latter just converges to e as the power of x increases although I'm not sure how to go about it (even knowing $lim (1+1/x)^x = e$, obviously, it seems non-trivial as I can't just ignore the "1+" but I'm not a mathematician and I am probably overlooking something trivial).
Kind of a peculiar first result, no? Is there any literature about these, or some insights?
For large $n$ and $x>1$, $(1+x^{-n})^{1+x^n}\approx(1+x^{-n})^{x^n}\approx e$, so the fixed point approximates $e$.
For the case $n=1$, a root $>1$ of $(1+x^{-1})^x=x^2/(x+1)$ will be a poor approximation for $e$, but will be a better approximation for the positive root of $x^2/(x+1)=e$, which is $\tfrac12(e+\sqrt{e^2+4e})\approx3.5$. This is still not a great approximation, as $x$ isn't large enough for $(1+x^{-1})^x$ to be very close to $e$.
We can iteratively improve this reasoning, but the result approximating $\pi$ looks like a coincidence. It's better to ask why the $n$ for which $\pi$ is the root is close to $1$.