how we know or prove that a function has a branch infinite
and how we study it?
You can prove it by taking the limit as the function goes to infinity at a point. For example if you want to prove that tan has two infinite branches, you would say:
$\displaystyle \lim_{x->\frac{\Pi}{2}^-}tan(x)=\infty$
$\displaystyle \lim_{x->\frac{\Pi}{2}^+}tan(x)=-\infty$
- Wolfram|Alpha