L'Hôpital's rule
L'Hôpital's rule is a fundamental principle in calculus used to resolve indeterminate limits, specifically those of the forms 0/0 or ∞/∞. When evaluating a limit that results in one of these indeterminate forms, L'Hôpital's rule allows mathematicians to find a determinate limit by taking the derivatives of the numerator and denominator of the function. This process may need to be repeated if the resulting limit is still indeterminate. Although named after the French mathematician Guillaume-Francois-Antoine L'Hôpital, evidence suggests that the rule was actually discovered by the Swiss mathematician Johann Bernoulli. L'Hôpital's contributions to mathematics include publishing the first book on differential calculus, titled *Analyse des Infiniment Petits*, in 1696, which significantly advanced the understanding of calculus. The recognition of L'Hôpital's rule has sparked debates over the true authorship of ideas within his works, as Bernoulli's contributions were substantial. Overall, L'Hôpital's rule remains a vital tool for students and professionals in mathematics, serving to simplify complex limit evaluations.
L'Hôpital's rule
L'Hôpital's rule (pronounced lopi-tall) is taught in all first-year calculus classes. It transforms an indeterminate limit into one that can be determined. A limit is the value approached by a function f(x) as x approaches a given value. An indeterminate limit is of the form 0/0 or ∞/∞(∞ represents infinity).
L'Hôpital's rule is named after the French mathematician Guillaume-Francois-Antoine L'Hôpital (also commonly known as L'Hospital). Because of this attribution, many people mistakenly credit L'Hôpital with the discovery when evidence suggests that it was actually made by the Swiss mathematician Johann Bernoulli. In 1696, L'Hôpital published the first book on differential calculus, Analyse des Infiniment Petits. L'Hôpital's rule appeared for the first time in this book.
Overview
Though it is likely that Johann Bernoulli discovered L'Hôpital's rule, Guillaume-Francois-Antoine L'Hôpital made significant contributions to the field of mathematics. L'Hôpital was born in Paris in 1661into a wealthy, prominent family. His passion for and abilities in mathematics became apparent when he was just a boy. It is believed that at the age of fifteen he solved a problem about the cycloid proposed by Pascal.
L'Hôpital joined the army, a common practice for wealthy men during this time. He rose to the rank of captain but had to resign because of nearsightedness. In Paris, he became a member of a group of amateur mathematicians who enjoyed a competitive rivalry in which they posed difficult questions to see who would be the first to come up with a solution. Scholars believe, however, that while L'Hôpital was the first to solve several problems, he was modest about his accomplishments—unlike his peers.
By 1686, a mathematician named Gottfried Wilhelm Leibniz published two papers on differential calculus. However, the papers were obscure and contained errors that made them difficult to comprehend. Leibniz left Paris after publishing the papers. Johann and Jakob Bernoulli, two brothers and Swiss mathematicians living in Paris, managed to make sense of the papers. L'Hôpital quickly gained a reputation for being one of the best mathematicians of his time.
L'Hôpital wanted to learn more about differential calculus. In 1691, he asked Johann Bernoulli, who was only twenty-four and had little money or social status, to help him understand the theories Leibniz had proposed. Bernoulli tutored L'Hôpital for four months both in Paris and at his country home in central France. A few years later, in 1695, L'Hôpital helped Bernoulli get a job at the University of Groningen in the Netherlands, securing his friend's financial future.
In 1696, L'Hôpital published the first book about differential calculus. Its full title is Analyse des Infiniment Petits, Pour l'intelligence des lignes courbes (Analysis of Infinitely Small Quantities for the Understanding of Curves). The book was a tremendous success, in part because it contained L'Hôpital's rule. As was the custom at this time, the book begins with definitions. After these, it discusses differential calculus. L'Hôpital's rule appears in a later chapter.
The Rule
L'Hôpital's rule works only with indeterminate limits, that is, limits of the form 0/0 or ∞/∞. The basic idea behind the rule is that if you try to determine a limit and it is indeterminate, you should try evaluating the limit of the function obtained by taking the derivatives of the numerator and denominator of the original function. (Note that this is not the same as the derivative of the original function, which requires the quotient rule for derivatives to evaluate.)
If you can get a determinate limit by this process, this answer is also the value of the original indeterminate limit. The rule may need to be applied more than once, if the resulting limit is again indeterminate. Algebraically, L'Hôpital's rule states:
Who Really Wrote the Book?
The question of who actually wrote Analyse des Infiniment Petits and made the discoveries within it has been the subject of much debate. L'Hôpital first published the book anonymously but in the introduction expressed his gratitude to Leibniz and the Bernoulli brothers.
When the book was published, Johann Bernoulli sent L'Hôpital a letter praising his accomplishment. After this, however, Bernoulli attempted to take credit for at least some of the ideas expressed in the text. In 1698, two years after the publication of the book, Bernoulli sent a letter to Leibniz claiming that L'Hôpital had copied his work. After L'Hôpital died in 1704, Bernoulli stepped up his claims, nearly accusing his friend of plagiarism. Scholars believe that Bernoulli may have waited until this time because he felt indebted to L'Hôpital while he was alive.
In the 1920s, Bernoulli's notes were published. Historians noted that his notes were extremely similar to parts of L'Hôpital's book. From the notes, they became convinced that Bernoulli actually discovered L'Hôpital's rule. In 1955, Bernoulli's early correspondence was published, which convinced those within the mathematics community that Bernoulli is the true author of Analyse des Infiniment Petits. Scholars theorize that the two men may have had a strange business arrangement in which L'Hôpital bought the rights to Bernoulli's ideas and could therefore publish them.
Before his death, L'Hôpital had completed a second book, Traité analytique des sections coniques et de leur usage pour la résolution des équations dans les problèmes tant déterminés qu'indéterminés. This book was published after his death in 1720. L'Hôpital had also planned to write a second volume of Analyse des Infiniment Petits about integral calculus but abandoned the effort when he heard that Leibniz was planning to write a book about the subject.
Bibliography
Broadwin, Judy. "The Immortal L'Hospital." AP Calculus. College Board. 2008. Web. 5 Dec. 2014. http://apcentral.collegeboard.com/apc/public/repository/ap08‗calc‗LHospital‗final.rev2.pdf
Robinson, Abraham. "L'hospital (L'hopital), Guillaume-FranCois-Antoine De (Marquis de Sainte-Mesme, Compte d'Entremont)." Encyclopedia.com. HighBeam Research, Inc. Web. 5 Dec. 2014. http://www.encyclopedia.com/doc/1G2-2830902606.html