I would like to approximate the function

$f(x)=frac{2x}{1-e^{-2x}}$

analytically for both small and large $x$.

But when I use the formula for the Taylor expansion, I run into the problem that the function and its derivative are not defined for $x=0$. How can I get around this problem?

For large $x$, my idea was to substitute $y:=1/x$ and then expand the function $g(y)=frac{2}{y(1-e^{-2x})}$ around $y=0$. However, here I run into the same problem as above.

How do you proceed in such a case?