L'Hôpital's rule is (essentially) a trick for finding limits of various indeterminant forms. It is probably due to Jean Bernoulli on whose lectures l'Hôpital's calculus textbook was based.
The easiest form to prove is one which assumes that both f and g are continuous and differentiable functions near x=a and both have value 0 at a. We seek
Standard techniques would give the form 0/0, which is indeterminant since examples can be found easily where the limit is 0, a number, or infinite (try letting f and g be powers and a=0 to come up with your own examples).
The intuition behind l'Hôpital's rule is that finding
is the same as finding
Now the mean value theorem applied to f and to g would say this is the same as
where a' and a'' are between x and a. The factors of (x-a) could then be canceled. If only a' and a'' were the same this would give l'Hôpital's rule. Since the mean value thoerem doesn't tell us that they are the same, we need a form of MVT which works with two functions instead of one.
Apply Rolle's theorem to the function
This function h is continuous on [a,b] because both f and g are, and is differentiable on (a,b) with
At the end points we get
By Rolle's theorem there is a c in (a,b) with h'(c)=0. This gives the needed point.
As before we think of
where is given by the Cauchy Mean Value theorem. Now since c is between x and a we know that as , we also have . Thus
This particular form of l'Hôpital's rule is not the most useful one. That is the form for limits as . The major use of that is to show that asymptotically grows slower than any power and grows faster than any power. The other major use of this form is to show that
a formula used for approximating compound interest by continuous compounding, one of the major uses of the exponential.
There are several standard pitfalls in using l'Hôpital's rule. The most obvious involve trying to use it on limits not of the form 0/0 or . The limit
which forms the basis for the derivation of the
derivative of sin(x) needs a geometric argument based on the definition
of radian measure and the trig functions as coordinates, rather than
using l'Hôpital because the limit is needed to find the derivative of the
function, so that derivative cannot be used in finding the limit.