1. Guillaume de l’Hôpital
a historical perspective to mathematical analysis

2. Overview Personal life Socioeconomic context
We will provide background from l’Hopital’s personal life, the socioeconomic context of 17th century France, and his contribution to mathematics relating to the topics covered in Advanced Calculus.
Socioeconomic context
Mathematical contributions

3. Personal life:

4. Military background high ranking military family
captain in a cavalry regiment

5. Influence from Johann Bernoulli
Congregation of the Oratory
Private lessons with Johann Bernoulli
“You have only to let me know your definite wishes, if I am to publish nothing more in my life, for I will follow them precisely and nothing more by me will be seen.”

6. First textbook on differential calculus
“I must own myself very much obliged to the labours of Messieurs Bernoulli, but particularly to those of the present Professor at Groningen. I have made free use of their discoveries, as well as those of Mr Leibniz, so that I frankly return to them whatever they please to claim as their own. I must here in justice own (as Mr Leibniz himself has done in 'Journal des Sçavans' for August 1694) that the learned Sir Isaac Newton likewise discovered something like the Calculus Differentialis ... But the method of Mr Leibniz is much more easy and expeditious, on account of the notation he uses, not to mention the wonderful assistance it affords on may occasions.”

7. Académie des Sciences
After the reorganization of the Academy of sciences by Louis XIV in 1699, l’Hôpital was given the status of honorary member.

8. Socioeconomic context:

9. The reign of Louis XIV and the Age of Reason 1643 1699 1715 1682
Beginning of the reign of Louis XIV
Foundation of the Royal Academy of Sciences
1715
1682
Death of the Sun King
Versailles became the permanent home for royalty and nobility

10. Political power Absolute monarchy Administrative and fiscal reforms
French influence in the world through commerce and trade
Old military aristocracy ceased to have a monopoly over senior military positions and rank
Revocation of the Edict of Nantes

11. Sciences in Versailles
Construction of The Versailles Castle
Education of Princes
Creation of Academies

12. Mathematical contribution: L’Hôpital’s Rule

13. Background Information
Leibniz’s (as well as Newton’s but to a smaller extent) invention of infinitesimal calculus was needed before L’Hôpital could do the work that led to the theorem that is known as L’Hôpital’s Rule. L’Hôpital mentions preferring Leibniz’s calculus for its simple and consistent notation. The same notation that is most commonly used today.
Leibniz Notation
y’=dy/dx
∫ for integration

14. Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes
Published in 1696
l’Hôpital’s Rule first appeared in Section IX of this book
Largely debated whether or not the math depicted in this book is l’Hôpital’s or Bernoulli’s, including l’Hôpital’s Rule.

15. L’Hôpital’s Rule
The rule states that for the differential functions f and g on an open interval. g’(x)≠0 for all x on the interval and the limit of f’(x)/g”’(x) exists.

16. Proof: Special Case
Since many functions have continuous derivatives, this case is worth mentioning.
Assume that f and g be continuously differentiable at the point c and f(c) = g(c) = 0. And g’(c) ≠ 0

17. Proof (cont.)
The proof follows from the difference-quotient definition of the derivative and the continuity of the derivatives at c. Since g’(c) ≠ 0 the limit is not indeterminate.