Generalizations of l'Hôpital's rule
Author:
Cheng Ming Lee
Journal:
Proc. Amer. Math. Soc. 66 (1977), 315320
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 realvalued 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 righthand side exists and that or .
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 B. S. Babcock, On properties of the approximate Peano derivatives, Trans. Amer. Math. Soc. 212 (1975), 279294. MR 0414803 (54:2895)
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 E. W. Hobson, The theory of functions of a real variable, 3rd ed., Cambridge Univ. Press, Cambridge, 1927, p. 359.
 [3]
 C. M. Lee, An approximate extension of CesàroPerron integrals, Bull. Inst. Math. Acad. Sinica Taiwan 4 (1976), 7382. MR 0412358 (54:484)
 [4]
 , On the approximate Peano derivatives, J. London Math. Soc. (2) 12 (1976), 475478. MR 0399378 (53:3222)
 [5]
 , An analogue of the theorem of HakeAlexandroffLooman, Fund. Math. (to appear). MR 0486362 (58:6109)
 [6]
 , Monotonicity theorems for approximate Peano derivatives and integrals, Real Exchange 1 (1976), 5262.
 [7]
 C. M. Lee and R. J. O'Malley, The second approximate derivative and the second approximate Peano derivative, Bull. Inst. Math. Acad. Sinica Taiwan 3 (1975), 193197. MR 0382563 (52:3446)
 [8]
 W. Rudin, Principles of mathematical analysis, McGrawHill, New York, 1953. MR 0055409 (14:1070c)
 [9]
 S. Saks, Theory of the integral, 2nd rev. ed., Warsaw, 1937.
 [10]
 Z. Zahorski, Sur la premiere dérivée, Trans. Amer. Math. Soc. 69 (1950), 154. MR 0037338 (12:247c)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197704539392
PII:
S 00029939(1977)04539392
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
