Power series is a good method because it is almost L'Hopital but without the need of using repeated derivatives.
@Polymerase: Taylor polynomial would be my approach .
here but I do not see how to complete the argument.
It is unnecessary mental encumbrance. It is implicit in power series and
asymptotic methods, but they are more generally useful.
The only place that I have encountered limits that have to be done using
L'Hopital's rule is in A-level and below maths homework which tell the student to use it.
(also if you meet limits in real life you often are interested in asymptotic behaviour as well as the limit)