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Let f and g be differentiable functions on the interval [a,b], with a < b.
Consider the following function:
h(x) = (f(x)  f(a)(g(b)  g(a))  (g(x)  g(a))(f(b)  f(a).
Then if we evaluate h at x = a and at x = b, we get:
h(a) = h(b) = 0.
So by Rolle's theorem there exists a point
in the interval (a, b), such that
.
Then we have
.
In particular, if g' is nonzero in the open interval (a,b) and if
,
we have, for some
in the interval (a,b):
This is called Cauchy's Mean Value Theorem.
In the special case that g(x) = x, so g'(x) = 1, this reduces to the ordinary mean value theorem.
Now consider the case that both f(a) and g(a) vanish and replace b by a variable x.
Then we have, provided
f(a) = g(a) = 0 and
in an interval around a, except possibly at x = a:
where ,
but x is near a and then
(which depends on x) is a point between x and a.
Now suppose
exists.
Then
also, so we get L'Hopital's rule:
It must be emphasized that this rule only has been shown to hold when both f(a) and g(a) vanish.
If now f and g are twice differentiable and f'(a) and g'(a) both vanish and if g''(x) is nonzero near x = a, except
possibly at x = a and then if
exists, we may apply L'Hopital's rule twice to give:
Finally if g and its first n1derivatives vanish at a, and if g and its first n derivatives do not vanish in an interval
around a, except possibly at a, then we have the iterated L'Hopital's rule:
Examples:
L'Hopital's rule is also valid for other cases, although we do not prove these:

;
a finite; this is the standard L'Hopital we proved above:
and applies to the case that
f(a) = g(a) = 0.

,
;
here
,
and
.

;
a finite; here
,
and
.

;
;
here
,
and
.
Examples:
Sometimes the calculation goes easier if we replace x by
.
Then a limit to
is replaced by a limit to 0^{+} and
a limit to
is replaced by a limit to 0^{} and viceversa.
Using this trick the last example reads:
Next we can handle some exponent limits by first taking logarithms and then using L'Hopital.
Consider
.
Let
,
then
and we have:
So
.
Finally, although again we do no prove this, L'Hopital can be used to show that limits do not exist.
For example:
which does not exist, since the numerator goes to 1 and the denominator goes to zero.
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George A. J. Sparling
20010111