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Guillaume François Antoine de l'Hôpital

Guillaume de l'Hôpital (1661-1704)was a French mathematician. His nameis firmly associated with l'Hôpital's rule for

calculating limits involving indeterminate forms0/0 and ∞/∞. Although the rule did notoriginate with l'Hôpital, it appeared in printfor the first time in his treatise on theinfinitesimal calculus, entitled Analyse des

Infiniment Petits pour l'Intelligence des LignesCourbes (Analysis of the Infinitely Small to Learn about Curved Lines) . This bookwas a first systematic exposition of differential calculus. Several editions andtranslations to other languages were published and it became a model forsubsequent treatments of calculus.

The Marquis de l’Hôpital, a French nobleman living by private meansserved briefly as a cavalry officer, but resigned because of his extreme

nearsightedness to devote his energies entirely to mathematics. In his time therecently invented calculus was fully understood only by Newton, Leibniz andthe Bernoulli brothers. In 1691-1692, when John Bernoulli spent over half ayear in Paris, he was generously compensated for giving the young Marquisprivate lessons on this powerful new method. In return for a monthlyallowance, Bernoulli was induced to continue the instruction by letter; theagreement was that he would communicate his future mathematicaldiscoveries exclusively to l’Hôpital. L’Hôpital eventually felt that he understoodthe material well enough to compose a proper textbook on it.

L’Hôpital’s Analyse des infiniment petits , published in 1696, contains anaccount of the differential calculus as conceived by Leibniz and learned fromBernoulli. In its preface l’Hôpital freely acknowledges his debt to the twomathematicians, saying: "I have made free use of their discoveries." Thesuccessive reprintings of the Analyse made the calculus known throughoutEurope. In 1730 it was translated into English, supplemented by the translatorwith work on the integral calculus; in tribute to Newton, the book’s derivative

notation was changed to the f luxional "dottage" of their English hero. L’Hôpitalis nowadays remembered in the name of his "0/0 and ∞/∞ rules " rules for

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