L'Hôpital's rule

In calculus (an area of mathematics), L'Hôpital's rule uses derivatives to determine otherwise hard to compute limitss. If you are trying to determine the limit of some quotient f(x)/g(x), and both the numerator and denominator approach 0 or infinity, then differentiate numerator and denominator and determine the limit of the quotient of the derivatives. If that limit exists, the rule states that it will be the same as the original limit.

That is,

if

Table of contents

1 Examples 2 Proof 3 Of interest

Examples

Here is a case of 0/0:

Here is a case of ∞/∞:

Sometimes, even limits which don't appear to be quotients can be handled with the same rule:

The rule is named after the 17th century French mathematician Guillaume François Antoine, Marquis de l'Hôpital (1661 - 1704), who published the rule in his book Analyse des infiniment petits pour l'intelligence des lignes courbes (1692), the first textbook to be written on the differential calculus.

Proof

The proof of L'Hôpital's rule depends on Cauchy's mean value theorem.

According to Cauchy's mean value theorem there is a constant in the interval such that:

Since , we can say that:

If we let , we get:

Therefore

Q.E.D.

Of interest

Although L'Hôpital's rule's rule is a powerful way of computing otherwise hard to compute limits, it is not always the easiest. Some special type of limits are actually easier to compute using the Taylor series expansion.

For example,