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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Generalizations of l'Hôpital's rule


Author: Cheng Ming Lee
Journal: Proc. Amer. Math. Soc. 66 (1977), 315-320
MSC: Primary 26A24
MathSciNet review: 0453939
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Abstract: An essential limit, similar to the concept of essential bounded functions, is defined and briefly discussed. Using the essential limit, l'Hôpital's rule is generalized to include the following theorem as a special case. Theorem. Let F, G be real-valued functions defined on the open interval (a, b). Suppose that the approximate derivatives $ {F'_{{\text{ap}}}}(x)$ and $ {G'_{{\text{ap}}}}(x)$ exist finitely, $ {G'_{{\text{ap}}}}(x) > 0$ for almost all x in (a, b), and the extreme approximate derivates of both F and G are finite nearly everywhere in (a, b). Then $ {\lim _{x \to a + }}[F(x)/G(x)] = {\text{ess}}\;{\lim _{x \to a + }}[{F'_{{\text{ap}}}}(x)/{G'_{{\text{ap}}}}(x)]$ provided that the essential limit in the right-hand side exists and that $ {\lim _{x \to a + }}F(x) = {\lim _{x \to a + }}G(x) = 0$ or $ {\lim _{x \to a + }}G(x) = - \infty $.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0453939-2
PII: S 0002-9939(1977)0453939-2
Keywords: Essential limit, ordinary limit, approximate limit, approximate Peano derivatives and derivates, generalized absolutely continuous functions, closed monotone functions, monotonicity theorem, l'Hôpital's rule
Article copyright: © Copyright 1977 American Mathematical Society