On this page
If you have solved a limit like (x² − 4)/(x − 2) as x → 2 by differentiating top and bottom, you have used l'Hôpital's rule. The man it is named after did not discover it, and his own introduction admits a large debt to others. He did something else that mattered: he wrote the first textbook of differential calculus. Our post on Johann Bernoulli tells the story from his teacher's side. This one is about l'Hôpital, his book and how to use the rule properly.
Who was l'Hôpital?
The MacTutor biography gives his dates as 1661 to 2 February 1704, both in Paris. He came from a noble military family and served as a cavalry captain. His contemporary Bernard de Fontenelle wrote that he studied geometry "even in his tent", partly to hide it, since some nobles thought learning beneath them. At about fifteen he solved a hard problem on the cycloid that Pascal had posed. He left the army because of poor eyesight.
In late 1691 the 24-year-old Johann Bernoulli arrived in Paris, knowing more about Leibniz's new calculus than anyone there. Bernoulli lectured to a circle of Paris mathematicians and then gave l'Hôpital private lessons at his country estate.
What was the arrangement with Bernoulli?
On 17 March 1694 l'Hôpital wrote offering Bernoulli a retainer of 300 pounds a year. In MacTutor's translation he asked Bernoulli "to communicate to me your discoveries, at the same time asking you not to disclose any of them to others." Bernoulli accepted.
In 1696 l'Hôpital published *Analyse des infiniment petits pour l'intelligence des lignes courbes* (*Infinitesimal analysis for the understanding of curved lines*). In the introduction he wrote: "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." MacTutor notes that only in 1921 did a manuscript copy of Bernoulli's course turn up, showing how closely the book followed it.
What was in the book?
MacTutor describes its structure. It opens with two definitions and two axioms: a quantity changed by an infinitely small amount can be treated as unchanged, and a curve can be treated as a polygon with infinitely many tiny sides. It then covers tangents, maxima and minima, points of inflection, curvature and higher-order differentials. The rule for a fraction whose top and bottom both tend to zero is in Chapter 9.
MacTutor credits l'Hôpital with understanding new mathematics quickly and writing "with great clarity". The book stayed in use for decades, with new editions until 1781, and became a model for later calculus textbooks. He also asked a good question of his own: in a 1695 letter to Leibniz he wondered whether a half-order derivative, dⁿy/dxⁿ with n = 1/2, could mean anything.
Worked examples
Example 1. Limit of (x² − 4)/(x − 2) as x → 2. Both top and bottom are 0 at x = 2. Differentiate each: 2x and 1. At x = 2 that gives 4/1 = 4. Factorising gives the same: (x − 2)(x + 2)/(x − 2) = x + 2 → 4.
Example 2. Limit of (1 − cos x)/x² as x → 0. Both are 0 at x = 0. Differentiate: sin x/(2x), still 0/0. Differentiate again: cos x/2, which tends to 1/2. So the limit is 1/2. Check with x = 0.1: 1 − cos 0.1 = 0.0049958, divided by 0.01 is 0.49958.
Example 3. Limit of (eˣ − 1)/x as x → 0. Differentiate: eˣ/1 → 1.
When does the rule go wrong?
Using it when the limit is not 0/0. For (x + 1)/(x + 2) as x → 0, the answer is simply 1/2. Differentiating would give 1/1 = 1, which is wrong.
Using the quotient rule. The rule differentiates top and bottom separately. It does not differentiate the fraction.
Circular reasoning. Using the rule on sin x/x assumes the derivative of sin x, which is itself usually proved from that very limit. Know the standard limits independently.
Stopping at another 0/0. If the new fraction is still 0/0, as in example 2, apply the rule again or simplify.
Our guide to limits and continuity for JEE has more practice with standard limits. If you keep reaching for the rule when a factorisation would do, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer.
What can a student take from l'Hôpital?
He was not the discoverer, and the arrangement he made is nothing to admire. But he saw that the new calculus needed a clear, ordered book, and he wrote one that taught people for nearly a century. That is a real skill. You can practise it: after you learn a topic, write a one-page explanation in your own words, with definitions first and one example per idea. Gaps in your understanding show up fast.
Try it now: find the limit of (x³ − 8)/(x² − 4) as x → 2 twice, once by factorising and once with the rule, and check that both give 3.
