About the Proof of the L’Hôpital's Rule
Keywords:
Locally defined, L’Hopital’s rule, Limit.Abstract
The recent proofs of L’Hopital’s rule require the continuity of the derivatives of the functions in an interval. In this work, a proof is given so that it does not require the existence of the derivatives of the functions in any other number than the number where the limit of its ratio is calculated, more precisely the L’Hopital rule extends to locally defined and derivable functions in the number where the limit of the ratio of the functions is calculated.
References
[1] R. Estrada & M. Pavlovi. “L’hôpital’s monotone rule, Gromov’s theorem, and operations that preserve the monotonicity of quotients”, Publ. Inst. Math. (Beograd) (N.S.) ,101,11–24, 2017.
[2] H. Struve & I. Witzke, “ Die Regel von l’Hôpital” , Elem. Math.,69, 118–129, july 2017
[3] S. Zlobec,”L’Hôpital’s rule without derivatives”. Math. Commun,67, 665–672, 2012
[4] N. Martins & D. Torres, “L’Hôpital-type rules for monotonicity with application to quantum calculus”. Int. J. Math. Comput. 10, 99–106 ,2011.
[5] A. V. Shishkina. “On the inversion of the l’Hôpital rule for functions holomorphic in the ball”, Izv. Vyssh. Uchebn. Zaved. Mat.,50, 76–82,2006..
[6] A. V. Shishkina. “On the inversion of the l’Hôpital rule for analytic functions”, Sibirsk. Mat. Zh., 46, 957–962 2005.
[7] Popa, Dumitru, “On the vector form of the Lagrange formula, the Darboux property and l’Hôpital’s rule”, Real Anal. Exchange,25,787–793,1999
[8] Takeuchi.. Yu “L’Hôpital’s rule for series”,Bol. Mat., 1,17–33,1995.
[9] I. Muntean. “L,’Hôpital’s rules with extreme limits” in Seminar on Mathematical Analysis, 1993, 11–28.
[10] M.Vianello. “ A generalization of l’Hôpital’s rule via absolute continuity and Banach modules”. Real Anal. Exchange, 18, 557–567.1993.
[11] Y.X. Tian, “ L’Hôpital rules for conjugate analytic functions”, Sichuan Shifan Daxue Xuebao Ziran Kexue Ban 16,53–56,1993.
[12] R. Spigler & M. Vianello. “Abstract Abstract versions of L'Hôpital's rule for holomorphic functions in the framework of complex B-modules” J. Math. Anal. Appl,17–28,1993.
[13] A. L. Durán. “The converse of de l’Hôpital’s rule”. Cienc. Tecn. ,16, 111–119. 1992.
[14] Hartig, Donald L’Hôpital’s rule via integration. Amer. Math. Monthly 98 (1991), no. 2, 156–157.
[15] Shishkina, A. V. On the inversion of the l’Hôpital rule for functions holomorphic in the ball. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 2006, no. 6, 78--84; translation in Russian Math. (Iz. VUZ) 50 (2006), no. 6, 76–82 (2007) (Reviewer: Sergey Ivashkovich) 32A10.
[16] Popa, Dumitru On the vector form of the Lagrange formula, the Darboux property and l’Hôpital’s rule.
[17] Takeuchi, Yu L’Hôpital’s rule for series. (Spanish) Bol. Mat. 1/2 (1995), no. 2-1, 17–33.
[18] Muntean, Ioan L’Hôpital’s rules with extreme limits. Seminar on Mathematical Analysis (Cluj-Napoca, 1992–1993), 11–28, Preprint, 93-7, " Babe-Bolyai” Univ., Cluj-Napoca, 1993.
[19] Spigler, Renato; Vianello, Marco Abstract versions of L’Hôpital’s rule for holomorphic functions in the framework of complex B-modules. J. Math. Anal. Appl. 180 (1993), no. 1, 17–28.
[20] D. Hartig. L’Hôpital’s rule via integration”, Amer. Math. Monthly, 98, 156–157. 199.
[21] J. Aczél. “Functional equations and L’Hôpital’s rule in an exact Poisson derivation”, Amer. Math. Monthly, 97, 423–426. 1990.
22] G. Szabó, "A note on the L’Hôpital’s rule", Elem. Math. 44 , 150–153,1989.
[23] R. Výborný & R. Nester, "L’Hôpital’s rule, a counterexample". Elem. Math. 44, 116–121,1989.
[24]W. P Cooke, "The Teaching of Mathematics: L’Hopital’s Rule in a Poisson Derivation". Amer. Math. Monthly 95, 253–254.1988.
[25] X. C. Huang," A discrete L’Hôpital’s rule",College Math. J., 19, 321–329,1988.
[2] H. Struve & I. Witzke, “ Die Regel von l’Hôpital” , Elem. Math.,69, 118–129, july 2017
[3] S. Zlobec,”L’Hôpital’s rule without derivatives”. Math. Commun,67, 665–672, 2012
[4] N. Martins & D. Torres, “L’Hôpital-type rules for monotonicity with application to quantum calculus”. Int. J. Math. Comput. 10, 99–106 ,2011.
[5] A. V. Shishkina. “On the inversion of the l’Hôpital rule for functions holomorphic in the ball”, Izv. Vyssh. Uchebn. Zaved. Mat.,50, 76–82,2006..
[6] A. V. Shishkina. “On the inversion of the l’Hôpital rule for analytic functions”, Sibirsk. Mat. Zh., 46, 957–962 2005.
[7] Popa, Dumitru, “On the vector form of the Lagrange formula, the Darboux property and l’Hôpital’s rule”, Real Anal. Exchange,25,787–793,1999
[8] Takeuchi.. Yu “L’Hôpital’s rule for series”,Bol. Mat., 1,17–33,1995.
[9] I. Muntean. “L,’Hôpital’s rules with extreme limits” in Seminar on Mathematical Analysis, 1993, 11–28.
[10] M.Vianello. “ A generalization of l’Hôpital’s rule via absolute continuity and Banach modules”. Real Anal. Exchange, 18, 557–567.1993.
[11] Y.X. Tian, “ L’Hôpital rules for conjugate analytic functions”, Sichuan Shifan Daxue Xuebao Ziran Kexue Ban 16,53–56,1993.
[12] R. Spigler & M. Vianello. “Abstract Abstract versions of L'Hôpital's rule for holomorphic functions in the framework of complex B-modules” J. Math. Anal. Appl,17–28,1993.
[13] A. L. Durán. “The converse of de l’Hôpital’s rule”. Cienc. Tecn. ,16, 111–119. 1992.
[14] Hartig, Donald L’Hôpital’s rule via integration. Amer. Math. Monthly 98 (1991), no. 2, 156–157.
[15] Shishkina, A. V. On the inversion of the l’Hôpital rule for functions holomorphic in the ball. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 2006, no. 6, 78--84; translation in Russian Math. (Iz. VUZ) 50 (2006), no. 6, 76–82 (2007) (Reviewer: Sergey Ivashkovich) 32A10.
[16] Popa, Dumitru On the vector form of the Lagrange formula, the Darboux property and l’Hôpital’s rule.
[17] Takeuchi, Yu L’Hôpital’s rule for series. (Spanish) Bol. Mat. 1/2 (1995), no. 2-1, 17–33.
[18] Muntean, Ioan L’Hôpital’s rules with extreme limits. Seminar on Mathematical Analysis (Cluj-Napoca, 1992–1993), 11–28, Preprint, 93-7, " Babe-Bolyai” Univ., Cluj-Napoca, 1993.
[19] Spigler, Renato; Vianello, Marco Abstract versions of L’Hôpital’s rule for holomorphic functions in the framework of complex B-modules. J. Math. Anal. Appl. 180 (1993), no. 1, 17–28.
[20] D. Hartig. L’Hôpital’s rule via integration”, Amer. Math. Monthly, 98, 156–157. 199.
[21] J. Aczél. “Functional equations and L’Hôpital’s rule in an exact Poisson derivation”, Amer. Math. Monthly, 97, 423–426. 1990.
22] G. Szabó, "A note on the L’Hôpital’s rule", Elem. Math. 44 , 150–153,1989.
[23] R. Výborný & R. Nester, "L’Hôpital’s rule, a counterexample". Elem. Math. 44, 116–121,1989.
[24]W. P Cooke, "The Teaching of Mathematics: L’Hopital’s Rule in a Poisson Derivation". Amer. Math. Monthly 95, 253–254.1988.
[25] X. C. Huang," A discrete L’Hôpital’s rule",College Math. J., 19, 321–329,1988.
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Published
2018-04-01
How to Cite
Corona-Corona, G. (2018). About the Proof of the L’Hôpital’s Rule. American Scientific Research Journal for Engineering, Technology, and Sciences, 41(1), 240–245. Retrieved from https://asrjetsjournal.org/index.php/American_Scientific_Journal/article/view/3982
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