Limits of real bivariate rational functions

TL;DR

This paper establishes conditions for the existence of limits of bivariate rational functions at points, characterizes the set of possible limits when the denominator has an isolated zero, and provides an algorithm using Puiseux expansions.

Contribution

It introduces a geometric approach using Puiseux expansions to determine and compute limits of bivariate rational functions at points, including the set of all possible limits.

Findings

Conditions for limit existence are characterized.

The set of possible limits forms a closed interval.

An effective algorithm for limit verification and computation is proposed.

Abstract

Given two nonzero polynomials $f, g \in\mathbb R[x,y]$ and a point $(a, b) \in \mathbb{R}^2,$ we give some necessary and sufficient conditions for the existence of the limit $\displaystyle \lim_{(x, y) \to (a, b)} \frac{f(x, y)}{g(x, y)}.$ We also show that, if the denominator $g$ has an isolated zero at the given point $(a, b),$ then the set of possible limits of $\displaystyle \lim_{(x, y) \to (a, b)} \frac{f(x, y)}{g(x, y)}$ is a closed interval in $\overline{\mathbb{R}}$ and can be explicitly determined. As an application, we propose an effective algorithm to verify the existence of the limit and compute the limit (if it exists). Our approach is geometric and is based on Puiseux expansions.