Guillaume De l'Hôpital

By: Alex Asay

http://www.gap-system.org/~history/Posters2/De_L%27Hopital.html

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 15Solving 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 yearsBernoulli 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

http://www.camotruck.net/rollins/math/lhopital-h.gif

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, 1704.

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. http://www.bookrags.com/biography/guillaume-francois-antoine-lhospital-wom/

l'Hôpital's rule. May 15, 2009. <http://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule>

Kensington Intermediate Senior High. <http://www.edu.pe.ca/kish/Grassroots/math/guillaum.htm>

Additional Cited Sources JOC/EFR. 2008. < http://www-

history.mcs.standrews.ac.uk/Biographies/De_L'Hopital.html>

Dictionary.com, LLC. <http://dictionary.reference.com/browse/axioms>