## Monday Morning Math: The L’Hôpitals

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Good morning! The math tidbit for today is a two-for-one special: the L’Hôpitals, who lived in France in the 1600s.  But we’ll start with the phrase that my brain jumps to when I see L’Hôpital, which is L’Hôpital’s Rule (also called L’Hospital’s Rule).  It’s about limits, so it shows up in Calculus.  Essentially, if you have a limit as $x\to a$ of a fraction where both the numerator and denominator individually are approaching 0 or where both the numerator and denominator individually are approaching $\pm\infty$, then:

provided that second limit actually exists.  So, for example,

$\lim_{x\to 0} \frac{\sin(x)}{x}=\lim_{x\to 0} \frac{\cos(x)}{1}=1$.

Huzzah!

This rule was named after Guillaume François Antoine de l’Hôpital, where that last name was spelled different ways even in his lifetime, even by him.  He learned calculus by correspondence with Johann Bernoulli, who was busy figuring out calculus himself at that time, since the subject was only a few decades old.  In 1696 l’Hôpital published what is considered to be the first Calculus book: Analyse des infiniment petits pour l’intelligence des lignes courbes.  He thanked several people in the introduction, including Johann Bernoulli, but it wasn’t clear at that time how much of the book was really after l’Hôpital’s own work (none?) and how much was based on Bernoulli’s notes (all?).  Apparently Bernoulli was fine with l’Hôpital publishing the book, possibly because of the money l’Hôpital paid him, possibly because he was happy just to have these still-new ideas disseminated. L’Hôpital died in 1704 when he was about 43 years old.

A lot of that information  comes from the MacTutor biography, which also states, “L’Hôpital married Marie-Charlotte de Romilley de La Chesnelaye; they had one son and three daughters.”   The English Wikipedia page adds that his wife was “also a mathematician and a member of the nobility, and inheritor of large estates in Brittany” with a link to a page for her – in French – from a French biography that indicates that she worked in Geometry and Algebra and lived from 1671-1737.  The only other site online that mentions her is this dictionary, also in French, which indicates that she helped with the printing of the aforementioned Calculus book and impressed another math professor, Monsieur de la Montre, with her knowledge of Euclid.  So was she too involved in the creation of that first Calculus book but not mentioned?  It sounds like it, though the extent of her involvement is unclear.

And we’ll leave on that uncertain note.  Any new information would of course be welcome.

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