Year: 2022
Type: preprint
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.
Source: arXiv (Cornell University)
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4226487216