Not that the above is not quite proving the full 0/0 case of L'Hopital, it is proving the weaker assertion that if f,g are diffble on an interval containing c and g'(c) is nonzero then f(x)/g(x)-> f'(c)/g'(c).
The usual formulation of L'Hopital's is that if f,g are diffble on a interval except possibly at an interior point c, and if f'(x)/g'(x)-> L as x->c, then f(x)/g(x) must also tend to L.
This is a bit more general as can be seen by functions involving things like square roots, which are not differentiable at 0, but you could still look at the limiting behaviour of the derivative as you approach 0.
The weaker formulation would probably suffice for pretty much any limit you would need to compute in MX2 though, just something to keep in mind.