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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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* ${\lim _{x \to a + }}G(x) = - \infty$.

- Bruce S. Babcock,
*On properties of the approximate Peano derivatives*, Trans. Amer. Math. Soc.**212**(1975), 279–294. MR**414803**, DOI https://doi.org/10.1090/S0002-9947-1975-0414803-0
E. W. Hobson, - Cheng Ming Lee,
*An approximate extension of Cesàro-Perron integrals*, Bull. Inst. Math. Acad. Sinica**4**(1976), no. 1, 73–82. MR**412358** - Cheng Ming Lee,
*On the approximate Peano derivatives*, J. London Math. Soc. (2)**12**(1975/76), no. 4, 475–478. MR**399378**, DOI https://doi.org/10.1112/jlms/s2-12.4.475 - Cheng Ming Lee,
*An analogue of the theorem of Hake-Alexandroff-Looman*, Fund. Math.**100**(1978), no. 1, 69–74. MR**486362**, DOI https://doi.org/10.4064/fm-100-1-69-74
---, - C. M. Lee and Richard J. O’Malley,
*The second approximate derivative and the second approximate Peano derivative*, Bull. Inst. Math. Acad. Sinica**3**(1975), no. 2, 193–197. MR**382563** - Walter Rudin,
*Principles of mathematical analysis*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. MR**0055409**
S. Saks, - Z. Zahorski,
*Sur la première dérivée*, Trans. Amer. Math. Soc.**69**(1950), 1–54 (French). MR**37338**, DOI https://doi.org/10.1090/S0002-9947-1950-0037338-9

*The theory of functions of a real variable*, 3rd ed., Cambridge Univ. Press, Cambridge, 1927, p. 359.

*Monotonicity theorems for approximate Peano derivatives and integrals*, Real Exchange

**1**(1976), 52-62.

*Theory of the integral*, 2nd rev. ed., Warsaw, 1937.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
26A24

Retrieve articles in all journals with MSC: 26A24

Additional Information

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