Math Help - L'hopital's Rule

In this section of Math Help online, we discuss L'hopital's Rule which nicely follows Rolle's Theorem and Mean Value Theorem.

What is L'Hospital's Rule?

L'Hopital's Rule or L'Hospital's Rule (sometimes just L'Hopital) is a rule permitting the evaluation of the limit of an indeterminate quotient of functions as the quotient of the limits of their derivatives. For example, for sin x,

 is an indeterminate of the form 0/0 but it can be evaluated as

Definition of L'Hopital's Rule

L'Hospital's Rule is an application of the Mean Value Theorem and lies in the evaluation of

where f(a) = 0 and g(a) = 0.

The L'Hopital Rule states that if the

is an indeterminate form (such as 0/0), then we can differentiate the numerator and the denominator separately and arrive at an expression that has the same limit as the original problem. Thus,

Who invented L'Hopital's Rule?

L'Hopital's Rule was named after the French Math analyst and geometer, Guillaume Francois Antoine de l'Hopital, Marquis de St Mesme (1661 - 1704). Guillaume Francois Antoine de l'Hopital was the author of the first textbook on differential calculus, but Guillaume Francois Antoine de l'Hopital is believed to have bought the rights to this rule from its discoverer.