Generalizations of l'Hôpital's rule

Author:
Cheng Ming Lee

Journal:
Proc. Amer. Math. Soc. **66** (1977), 315-320

MSC:
Primary 26A24

DOI:
https://doi.org/10.1090/S0002-9939-1977-0453939-2

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* *and* *exist finitely*, *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* *provided that the essential limit in the right-hand side exists and that* *or* .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1977-0453939-2

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