Limits of quotients of real analytic functions in two variables

TL;DR

This paper establishes necessary and sufficient conditions for the existence of limits of quotients of real analytic functions in two variables, and provides an algorithm implemented in MAPLE to determine and compute these limits.

Contribution

It introduces new criteria for limit existence of quotients of real analytic functions and presents a more powerful algorithm than existing MAPLE tools.

Findings

Provides necessary and sufficient conditions for limit existence.

Develops an algorithm implemented in MAPLE 12.

Uses Hensel's Lemma and Puiseux series theory.

Abstract

Necessary and sufficient conditions for the existence of limits of the form {equation*} \lim_{(x,y)\rightarrow (a,b)}\frac{f(x,y)}{g(x,y)} {equation*} are given, under the hipothesis that $f$ and $g$ are real analytic functions near the point $(a,b)$, and $g$ has an isolated zero at $(a,b)$. An algorithm (implemented in MAPLE 12) is also provided. This algorithm determines the existence of the limit, and computes it in case it exists. It is shown to be more powerful than the one found in the latest versions of MAPLE. The main tools used throughout are Hensel's Lemma and the theory of Puiseux series.