analysis – Taylor expansion of a function

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?