It's explained in the Zero over Zero proof: L'Hôpital's rule - Wikipedia, the free encyclopedia
A common way to proof De l'Hôpital's rule (for both function limits going to zero as x approaches point a) consists of "extending" the functions so they are defined in point a. Intuitively I can see why this extension is okay, but mathematically we're introducing and using a new assumption here - why is this allowed?
It's explained in the Zero over Zero proof: L'Hôpital's rule - Wikipedia, the free encyclopedia