L'Hopital's Rule is used to differentiate a quotient of two functions
at a point where f and g are both zero. 0 over 0 is undefined so it might be thought the derivative of f over g does not exist. Of ten however the derivative does exist and we can evaluate it using L'Hopital's Rule:
If
and
then
if
and
are not both zero.
L'Hopital's rule can be used repeatedly so if
are all zero then
Proof: A simple proof can be constructed using Taylor series for
and
around the point
so
Now let
and all the
and higher order terms vanish. Hence
Example: Differentiate
at
so L'Hopital' Rule applies.
