Guillaume De l'Hôpital By: Alex Asay

Background Information  Born in Paris in 1661  Parents were Anne-Alexandre de l'Hospital and Elizabeth Gobelin  Became the nobleman of St. Mesme  Great at math while growing up  Stunned math geniuses by the time he was 15 Solving an “impossible” problem

Other Background Information  Served as a cavalry officer in the French Army  Had to resign because of his near- sightedness  He hired Bernoulli in 1691 to teach him more about math, which he devoted himself

Spark of Fame  l'Hôpital became famous for writing a book Analyse des infiniment petits pour l'intelligence des lignes courbes (Analysis of the Infinitely Small to Learn about Curved Lines) Sparked controversy after five years Bernoulli claimed the ideas behind  This book contained “l'Hôpital’s Rule”

The Beginning  The first chapter jumps right into two definitions: Definition 1: “Variable quantities are those that increase or decrease continuously while a constant quantity remains the same while other vary.” Definition 2: “The infinitely small part by which a variable quantity increases or decreases continuously is called the differential of that quantity.”

The Beginning  After this, it moves onto two axioms (commonly accepted principles and rules): Axiom 1: “Grant that two quantities whose difference is an infinitely small quantity may be taken (or used) indifferently for each other; or (what is the same thing) that a quantity which is increased or decreased only by an infinitesimally small quantity may be considered as remaining the same.” Axiom 2: “Grant that a curved line may be considered as the assemblage of an infinite number of infinitely small straight lines; or (what is the same thing) as a polygon with an infinite number of sides, each of infinitely small length such that the angle between adjacent lines determines the curvature of the curve.”

“l'Hôpital’s Rule”  It states a method for finding the limit of a rational function that has both a denominator and numerator of zero at a point.  If the limit of f(x) as x approaches c which equals the limit of g(x) as x approaches c which equals 0 or the limit of g(x) equals plus or minus infinity and the limit of f’(x) divided by g’(x) as x approaches c exists, then the limit of f(x)/g(x) as x approaches c equals the limit of f’(x)/g’(x) as x approaches c.

Requirements of the Limit  In order for “l'Hôpital’s Rule” to exist, the limit of f’(x)/g’(x) as x approaches c must exist

The Rule in Numerical Form

Results of the Controversy  l'Hôpital was wrong!  In 1921, an early Bernoulli manuscript was discovered revealing very similar traits to the items discussed in l'Hôpital’s book  On March 17, 1694, he sent a letter to Bernoulli asking him if he would stay quiet about the dilemma if he would get paid. Basically, l'Hôpital bribed Bernoulli to shut up!!! Bernoulli accepted

Additional Fame  Found the solution to the brachystochrone (the curve of the fastest decent in a gravitational field of a weighted particle that is moving between two points)  The problem was solved by others including Newton, Leibnitz, and Jacob Bernoulli

Additional Fame  He also published a rule on analytical conics in 1707 that was considered to be the standard reference for that topic for about 100 years.

The l'Hôpital Legacy  l'Hôpital died in Paris on February 2,  It is believed that he was a generous, modest, and charismatic man  He had three kids with his wife Marie- Charlotte de Romilley de La Chesnelaye.

Cited Sources  World of Mathematics. World of Mathematics on Guillaume Francois Antoine L'Hospital. Guillaume Francois Antoine L'Hospital Biography. francois-antoine-lhospital-wom/ francois-antoine-lhospital-wom/  l'Hôpital's rule. May 15, rule  Kensington Intermediate Senior High. uillaum.htm

Additional Cited Sources  JOC/EFR history.mcs.standrews.ac.uk/Biographie s/De_L'Hopital.html  Dictionary.com, LLC. axioms