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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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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