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#
