In mathematical jargon, a limit describes a function's behavior as its argument approaches some value. There are rules for finding every limit, even if it's for a quotient, and even if that quotient has one term that goes to zero or infinity. But if both numerator and denominator are indeterminate, then it becomes a race of sorts.

Sometimes it's clear-cut, as when one term approaching the extreme value fastest determines the entire expression's limit. Other times, when numerator and denominator approach an extreme value at a similar rate, there's a reconciliation and the limit is some “compromise” value — determined by L'Hôspital's rule.

Ratio race

Named after the 17th-century mathemetician Marquis de L'Hôspital Guillaume François Antoine, L'Hôspital's rule can return the limit of such quotients by making them determinate. Whether it's % or ·/·, L'Hôspital's rule states that differentiation of both numerator and denominator does not change a quotient's limit, thus simplifying the solution.

A nod to teachers

The Marquis de L'Hôspital wrote the first known book on differential calculus, L'Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes, in 1696. Through mention of lectures on the % indeterminate form, he revealed that his tutor Johann Bernoulli was actually the first to discover the validity of the rule we explore here.

Going nowhere fast

Although L'Hôspital's rule is a powerful tool for computing otherwise elusive limits, it's not always the easiest to use. In fact, some limits are best derived leveraging other special cases of the Mean Value Theorem — the Taylor series, for example. That said, some ratios that don't initially appear to qualify for treatment by L'Hospital's rule can sometimes be manipulated into a form that does.

Prove you mean it

Supporting graphical proofs, the most thorough proof uses Cauchy's Mean Value Theorem: On any section of a smooth curve, there is a point at which the slope equals the section's average gradient.

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