New Approaches for Evaluation Indeterminate Limits for Multivariable Functions in Undergraduate Mathematics Courses

Abstract

Zero divided by zero is one of the most important indeterminate forms obtained when evaluating limits for single variable functions and series in calculus education. Well-known method; L'Hôpital rule and its generalized form have been employed to simplify and resolve the indeterminate form such that zero divided by zero in terms of quotients of their derivatives for single variable functions as well as for multivariable functions. Nevertheless, L' Hôpital rule is impractical for the indeterminate limit forms of two variable functions in some cases such that isolated and nonisolated singularities, requirement of application of L'Hôpital rule more than once and complexity of taking derivative for some multivariable functions. So L'Hôpital rule cannot be preferred due to these reasons. By considering all these facts, new approaches including Finite Differences such as Central (CFD), Forward (FFD), Backward (BFD), High Accurate Central (HACFD), High Accurate Forward (HAFFD), High Accurate Backward (HABFD) methods, and Richardson Extrapolation method are presented that provide efficient ways to solve these limits instead of using L' Hôpital rule. Error analysis is also performed. All these methods are compared with each other in terms of accuracy and computational efficiency. It is observed that these approaches will be good alternatives instead of L'Hôpital rule for indeterminate form of two variable functions in calculus courses for both instructors and their students. Numerical examples are presented for this purpose.

Keywords

• Zero divided by zero
• L’Hôpital rule
• Central finite differences
• Forward finite differences
• Backward finite differences
• Richardson extrapolation method

References

1. Aczél, J. (1990). Functional equations and L’Hôpital’s rule in an exact Poisson derivation. American Mathematical Monthly, 97(5), 423–426.
2. Chapra, S. C., and Canale R. P, (2010). Numerical Methods for Engineers, Sixth Edition, Mc Graw Hill, New York.
3. Cooke, W. P. (1988). The Teaching of Mathematics: L’Hopital’s Rule in a Poisson Derivation. American Mathematical Monthly, 95(3), 253–254.
4. Corona-Corona, G. (2018). About the Proof of the L’Hôpital's Rule. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS), 41(1), 240-245.
5. Dinçkal, Ç. (2024). Additional chapter for evaluation indeterminate limits of functions and series in teaching mathematics for engineering education. International Journal of Engineering, Science and Technology, 16(4), 20-28.
6. Durán, A. L. (1992). The converse of de L’Hôpital’s rule. Ciencia e Tecnica, 16, 111-119.
7. Estrada, R., and Pavlović, M. (2017). L’Hôpital’s monotone rule, Gromov’s theorem, and operations that preserve the monotonicity of quotients. Publications de l'Institut Mathematique, 101(115), 11-24.
8. Fine, A. I., and Kass, S. (1966). Indeterminate forms for multi-place functions. Annales Polonici Mathematici, 1(18), 59-64.

9. Gordon, S. P. (2017). Visualizing and understanding L'hopital's rule. International Journal of Mathematical Education in Science and Technology, 48(7), 1096-1105.
10. Hartig, D. (1991). L’Hôpital’s rule via integration American. Mathematical Monthly, 98(2), 156–157.
11. Huang, X. C. (1988). A discrete L’ Hôpital’s rule. College Mathematics Journal, 19(4), 321–329. https://doi.org/10.1080/07468342.1988.11973132
12. Ivlev, V. V. (2013). Mathematical analysis: Multivariable functions, Moskow, IKAR in Russian.
13. Ivlev V. V., Shilin, I. A. (2014). On generalization of L’Hôpital’s rule for multivariable functions, arXiv: 1403.3006v1.
14. Lawlor, G. R. (2020). L’Hôpital’s rule for multivariable functions. American Mathematical Monthly, 127(8), 717-725 https://doi.org/10.1080/00029890.2020.1793635.
15. Muntean, L. (1993). L’Hôpital’s rules with extreme limits, Seminar on Mathematical Analysis (Cluj-Napoca, 1992–1993), Babe-Bolyai Univ., Cluj-Napoca 11–28, Preprint: 93-7
16. Popa, D. (1999). On the vector form of the Lagrange formula, the Darboux property and l’Hôpital’s rule, Real Analysis Exchange. 25(2): 787-794
17. Shishkina, A. V. (2007). On the inversion of the L’Hôpital rule for functions holomorphic in the ball. (Russian), Izvestiya Vyssh. Uchebn. Zaved. Mat. 2006, no. 6, 78-84; translation in Russian Mathematics (Iz. VUZ), 50(6): 76–82.
18. Spigler, R., and Vianello, M. (1993). Abstract Versions of L′ Hôpital′ s Rule for Holomorphic Functions in the Framework of Complex B-Modules. Journal of Mathematical Analysis and Applications, 180(1), 17-28.
19. Szabó, G. (1989). A note on the L’Hôpital’s rule. Elemente der Mathematik, 44, 150–153.
20. Takeuchi, Y. (1995). L’Hôpital’s rule for series. Bol. Mat., 1/2. 2(1), 17-33.
21. Tian, Y. X. (1993). L’Hôpital rules for conjugate analytic functions. Sichuan Shifan Daxue Xuebao Ziran Kexue Ban, 16, 53-56.
22. Vianello, M. (1992). A generalization of l’Hôpital’s rule via absolute continuity and Banach modules. Real Analysis Exchange, 18(2), 1992/93, 557-56
23. Vyborny, R., and Nester, R. (1989). L'Hôpital's rule, a counterexample. Elemente der Mathematik, 44, 116-121.
24. Young, W. H. (1910). On indeterminate forms. Proceedings of the London Mathematical Society, 2(1), 40-76.
25. Zlobec, S. (2012). L'Hôpital's rule without derivatives. Mathematical Communications, 17(2), 665-672.