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

Year 2025, Volume: 2 Issue: 1, 56 - 74, 29.05.2025

Çiğdem Dinçkal

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

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