## Theorem 6.5.15: l Hospital Rules

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