Can $e^{\frac{1}{\ln x}}$ be simplified or roughly approximated

Question

Clearly $e^{\ln x} = x$.

Is there a simplification for $e^{\frac{1}{\ln x}}$

Answer 1

We have that for $x$ large or small since $\frac1{\log x} \to 0$

$$e^{\frac{1}{\ln x}} \approx1+\frac1{\ln x}$$

and more in general

$$e^{\frac{1}{\ln x}} \approx 1+\frac1{\ln x}+\frac12 \frac1{\ln^2 x}+\frac1{3!}\frac1{\ln^3 x}+\ldots+\frac1{n!}\frac1{\ln^n x}$$