l'Hopitals's Rule

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:

Ifandthenifandare not both zero.

L'Hopital's rule can be used repeatedly so ifare all zero then

Proof: A simple proof can be constructed using Taylor series forandaround the point

so

Now letand all theand higher order terms vanish. Hence

Example: Differentiateat

so L'Hopital' Rule applies.

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