Shinban Liu

04/20/07

Mat 266

Naala Brewer

Footnote 18

Writing Project 3.7: The Origins of L’Hopital’s Rule

Guillaume François Antoine l'Hopital was born in Paris, France in the year 1661. Initially, L’Hopital planned on a military career but because of his poor eyesight, he switched to mathematics. John Bernoulli on the other hand, was born in Bâle on August 7, 1667 and was raised in a family of great thinkers. However, Bernoulli was well known for being extremely jealous and vindictive. Once he attemped to substitute an incorrect solution of his own on the problem of isoperimetrical curves with another stolen from his brother James. Additionally, he expelled his son Daniel from his house for obtaining a prize from the French Academy that he expected to receive himself. After this incident, John Bernoulli took all the ideas of his son’s book Hydrodynamica and made his own book, making sure to predate it to receive the credit. Although L’Hopital himself was brilliant in math, in 1694 he hired this swiss nobleman to keep him updated on developments in the new subject of calculus and to solve problems that he had. Among these problems was that of limits of indeterminate forms.

As compensation for tutoring him in calculus, Bernoulli was paid 300 Francs per year and also entered into an agreement that gave L’Hopital the right to use Bernoulli’s discoveries as he pleased. In 1696, L’Hopital published the first known textbook on differential calculus, l'Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes—which literally means Analysis of the infinitely small to understand curves. This text was composed of works done mostly by Bernoulli and includes the lectures of Bernoulli, in which Bernoulli discusses the indeterminate rule for differentiation. In this rule, now deemed L’Hopital’s Rule, an indeterminate form of 0/0 can be converted into a determinate form by repeated differentiation. Today, this rule has been extended to include functions that are ∞/∞, ∞, 0 · ∞, 00, and ∞ - ∞.

In 1704, after L’Hopital’s death in Paris, Bernoulli revealed the deal to the world, claiming that many of the results in L’Hopital’s book were due to him. In 1922 texts were found that gave support for Bernoulli’s claim such as a letter by John Bernoulli that gives both the rule and its proof. However, the claim that L’Hopital stole Bernoulli’s proofs and rules to get credit is false because L’Hopital published his book anonymously, acknowledged Bernoulli’s help in the introduction, and never claimed to be responsible for the rule. All in all, the rule used today in calculus is still coined under L’Hopital and not under Benoulli who likely created it.