Theorem: l'Hospital's Rule

• If f and g are differentiable in a neighborhood of x = c, and f(c) = g(c) = 0, then
  • provided the limit on the right exists
• The same result holds for one-sided limits.
• If f and g are differentiable and f(x) = g(x) = -, then
  • , provided the last limit exists

Proof:

The first part can be proved easily, if the right hand limit equals f'(c) / g'(c): Since f(c) = g(c) = 0 we have

Taking the limit as x approaches c we get the first result. However, the actual result is somewhat more general, and we have to be slightly more careful. We will use a version of the Mean Value theorem:

Take any sequence {} converging to c from above. All assumptions of the generalized Mean Value theorem are satisfied (check !) on [c, ]. Therefore, for each n there exists a number in the interval (c, ) such that

Taking the limit as n approaches infinity will give the desired result for right-handed limits. The proof is similar for left handed limits and therefore for 'full' limits.

The proof of the last part of this theorem is left as an exercise.