Given $f(x)=x/(e^x-1),$ find the increasing and decreasing interval and sketch a graph

Question

I used limitation to prove that limit $x$ to $0$ $f(x)$ exists which is equal to $1,$ but I can't prove that whether $f(0)$ exist or not, I also found that limit $x$ to infinity $f(x)=0$ which $y=0$ is the horizontal asymptote and limit x to negative infinity $f(x)= \infty.$ Since I can't prove $f(0)$ exist or not, but when $f'(x)=0,$ I found that $x=0,$ $x=0$ is a critical point? I know the function is decreasing from negative infinity, but which interval should I write? (-infinity, infinity) or (-infinity,0)u(0,infinity)? or else?