L'Hospital's Rule

Let lim stand for the limit , , , , or , and suppose that lim and lim are both zero or are both . If

(1)

has a finite value or if the limit is , then

(2)

Historically, this result first appeared in l'Hospital's 1696 treatise, which was the first textbook on differential calculus. Within the book, l'Hospital thanks the Bernoulli brothers for their assistance and their discoveries. An earlier letter by John Bernoulli gives both the rule and its proof, so it seems likely that Bernoulli discovered the rule (Larson et al. 1999, p. 524).

Note that l'Hospital's name is commonly seen spelled both "l'Hospital" (e.g., Maurer 1981, p. 426; Arfken 1985, p. 310) and "l'Hôpital" (e.g., Maurer 1981, p. 426; Gray 1997, p. 529), the two being equivalent in French spelling.

L'Hospital's rule occasionally fails to yield useful results, as in the case of the function , illustrated above. Repeatedly applying the rule in this case gives expressions which oscillate and never converge,

			(3)
			(4)
			(5)
			(6)
			(7)

The actual limit is 1.

L'Hospital's rule must sometimes be applied with some care, since it holds only in the implicitly understood case that does not change sign infinitely often in a neighborhood of . For example, consider the limit with

			(8)
			(9)

as . While both and approach as , the limit of the ratio is bounded inside the interval , while the limit of approaches 0 (Boas 1986).

Another similar example is the limit with

			(10)
			(11)

as . While both and approach 0 as , the limit of the ratio is 0, while the limit is unbounded on the real line (Wilf 1966, Rickert 1968).