How can we solve a limit having an indetermination of the type 0/0 or infinity divided by infinity?

Sometimes it would be enough to factorise the polynomials (in case the limit we are trying to compute is writen in terms of polynomials) and eliminate common factors in the quotient. However, most of them can be solved by L`Hopital's roule, consisting in computing the derivative of the numerator and the derivative of the denominator and then doing the quotient.