06 September 2015

l'Hospital's rule, 1 idea with 75 cases

Short history

L' Hospital's rule is named after the French marquis de l'Hospital who, in 1696, published the first calculus book:

The title means Computing with infinitely small quantities to understand curved lines. Actually, the book is anonymous, and the name of its author appeared only in the posthumous second edition. In his preface, the author, though a capable mathematician, gives full credit to others, writing

i.e. For the rest I recognise that I owe much to the ideas of the Bernoulli gentlemen, particularly the younger who is now professor in Groningen. I have freely used their discoveries and those of M. Leibnis. I therefore grant them whatever they may want to claim, being content with anything they may leave me. The correspondence between Johann Bernoulli and de l'Hospital reveals that l' Hospital's rule was indeed the former's, but the latter is by no means to be blamed for the misnomer. (There was even a legal understanding between them, de l'Hospital paying for the right to use Bernoulli's results.)

Anyhow, here is the famous rule as it occurs in the book, followed by a not too trivial example (due to Bernoulli).

In case you need the figure, the whole book can be consulted here. Viewed with modern eyes, the calculus content under this geometrical description is very limited and is even weaker than the baby version in the baby-macho-extended hierarchy found here.

Baby l'Hospital (right-sided)

(Statement and proof in one.)

Elegant and useful, but rather trivial; most interesting limits occur in points adherent to but not in the domain of the functions.

Macho l'Hospital

There are five types of limits (right-sided, left-sided, twosided, to +∞, to -∞),  five indeterminate quotients (0/0 and, counting the signs, four types of ∞/∞), and three possible solutions (a real number, +∞, -∞). All in all, 75 cases for de l'Hospital to solve. Calculus texts fail to convey a common idea for this panoply; in any case, I was not aware of one. Technically, most proofs rely on Cauchy's generalized mean value theorem, which most of the time is included for this purpose only. Let's do better. The following elementary theorem expresses the common idea behind all this: bounds for f'/g' are essentially preserved for f/g.

And here are the 75 cases for L' Hospital's rule.