Proof of L'Hopital's rule
It appears you are trying to prove the converse of L'Hopital: If $\lim f(x)/g(x) = L,$ then $\lim f'(x)/g'(x) = L.$ The converse is false. Take $f(x) = x^2\sin (1/x), g(x) = x$ to see this (here $a=0$).
Proof of L'Hopital's rule
It appears you are trying to prove the converse of L'Hopital: If $\lim f(x)/g(x) = L,$ then $\lim f'(x)/g'(x) = L.$ The converse is false. Take $f(x) = x^2\sin (1/x), g(x) = x$ to see this (here $a=0$).