Limits of real bivariate rational functions

Abstract

Given two nonzero polynomials $f,g\in R[x,y]$ and a point $(a,b)\in R^2$, we give some necessary and sufficient conditions for the existence of the limit $\underset{(x,y)\to (a,b)}{\lim }\frac{f(x,y)}{g(x,y)}$. 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 $\underset{(x,y)\to (a,b)}{\lim }\frac{f(x,y)}{g(x,y)}$ is a closed interval in $\bar{R}$ 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.