L'hopital's rule is used primarily for finding the limit as x->a of a function of the form f(x)/g(x), when the limits of f and g at a are such that f(a)/g(a) results in an indeterminate form, such as 0/0 or oo/oo. In such cases, one can take the limit of the derivatives of those functions as x->a. Thus, one would calculate lim_(x->a) (f'(x))/(g'(x)), which will be equal to the limit of the initial function.
As an example of a function where this may be useful, consider the function sin(x)/x. In this case, f(x) = sin(x), g(x) = x. As x->0, sin(x) -> 0 and x -> 0. Thus,
lim_(x->0) sin(x)/x = 0/0 = ?
0/0 is an indeterminate form because we cannot precisely define what it is equal to.
However, by taking the derivatives, we find f'(x) = cos (x), g'(x) = 1. And thus...
lim_(x->0) sin(x)/x = lim_(x->0) cos(x)/1 = lim_(x->0) cos(x) = cos(0) = 1
