[Paper Review] On a generalization of l'Hopital's rule for multivariable functions

V. V. Ivlev, И А Шилин|arXiv (Cornell University)|Mar 9, 2014

Iterative Methods for Nonlinear Equations1 references3 citations

TL;DR

This paper proposes a multivariable generalization of l'Hôpital's rule for indeterminate forms of order one and higher, based on the behavior of first and second-order partial derivatives at a point. It establishes that the limit of a ratio of two functions vanishing at a point exists and equals a constant k if and only if the ratios of their corresponding partial derivatives are equal across all first-order (or higher-order) terms, providing a constructive method to build such indeterminate forms with a prescribed limit.

ABSTRACT

For students and their lecturers and instructors interested in the natural problem of a possible generalization of l'Hopital's rule for functions depending on two or more variables, we offer our approach. For instructors, we discuss the technique of constructing indeterminate forms at a given point and having a given double limit.

Motivation & Objective

• To address the lack of a multivariable analogue of l'Hôpital's rule in standard textbooks.
• To provide a systematic method for constructing multivariable indeterminate forms with a specified double limit.
• To establish necessary and sufficient conditions for the existence of a finite limit of a ratio of two functions vanishing at a point.
• To extend the classical rule to higher-order vanishing (order n) using higher-order partial derivatives.
• To support teaching and learning in mathematical analysis by offering a practical, classroom-ready framework for students and instructors.

Proposed method

• Define a function as infinitely small of order n at a point if its first through (n−1)th Fréchet derivatives vanish there, but the nth derivative does not.
• Introduce the concept of an indeterminate form of order n as the ratio of two functions both vanishing to order n at a point.
• For order one, derive a condition based on the equality of ratios of first-order partial derivatives: f_x/g_x = f_y/g_y = k.
• For higher orders, extend the condition to ratios of second-order partial derivatives (e.g., f_xx/g_xx = f_xy/g_xy = ... = k).
• Use Taylor expansions up to first or second order to analyze the asymptotic behavior of the ratio f/g near the point.
• Construct explicit examples by adding linear or quadratic correction terms to functions so that their partial derivative ratios match a desired limit k.

Experimental results

Research questions

• RQ1 Under what conditions does the limit of f(x,y)/g(x,y) exist when both f and g vanish at a point (x₀,y₀)?
• RQ2 Can l'Hôpital's rule be generalized to multivariable functions using partial derivatives?
• RQ3 How can one construct a multivariable indeterminate form with a prescribed finite limit?
• RQ4 What role do higher-order partial derivatives play in determining the limit of a ratio of vanishing functions?
• RQ5 Is there a systematic method to ensure the limit of f/g equals a given constant k by adjusting the functions' derivatives at the point?

Key findings

• The limit lim_{(x,y)→(x₀,y₀)} f(x,y)/g(x,y) exists and equals k if and only if f_x/g_x = f_y/g_y = k at (x₀,y₀), under the condition that both f and g are infinitely small of order one.
• For higher-order indeterminate forms (order n > 1), the limit exists and equals k if and only if all corresponding ratios of n-th order partial derivatives are equal to k.
• The paper provides explicit formulas for constructing correction terms (C₁, C₂ for first order; C₁*, C₂*, C₃* for second order) to modify functions so that the resulting ratio has a desired limit k.
• An example is constructed where f(x,y) = x²y + x + y and g(x,y) = x²y² + xy are adjusted by adding linear terms to yield a limit of 2 at (1,1).
• Another example shows that by adding quadratic terms, the first-order derivatives of numerator and denominator vanish at (1,1), yet the limit remains 2, confirming the rule for second-order forms.
• The method is validated through Taylor series expansions and is applied in teaching mathematical analysis to first-year university students.