You can find a simpler proof here:

https://web.ma.utexas.edu/users/m408n/CurrentWeb/LM4-4-11.php

with an extra assumption that lim_{x->a} f(x)/g(x) exists.

or another proof without this extra assumption here:

https://math.stackexchange.com/questions/147356/proof-of-the-lh%C3%B4pital-rule-for-frac-infty-infty

given separately by Américo Tavares (May 20 '12 at 17:12) and by SomeoneContinuous (May 21 '12 at 14:44)