Is there any literature surrounding functions of the form $f(x)=e^\frac{\ln(p(x))}{p(x)}=p(x)^{\frac{1}{p(x)}}$, where $p(x)$ is a polynomial? By graphical methods, it seems as though $f(x)$ has a tendency to become tangent to $p(x)$. It'd be nice to see more (possibly detailed) analysis of these functions.
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5It is $[p(x)]^\frac1{p(x)}$ which tends to $1$ as $p(x)\to \infty$. – user Jun 12 '23 at 21:10