Limits of quotients of bivariate real analytic functions

Abstract

Necessary and sufficient conditions for the existence of limits of the form lim (x,y)→ (a,b) f (x, y) g(x, y) 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.

Limits of quotients of polynomial functions in three variables

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A method for computing limits of quotients of real analytic functions in two variables was developed in [4]. In this article we generalize the results obtained in that paper to the case of quotients q = f (x, y, z)/g(x, y, z) of polynomial functions in three variables with rational coefficients.

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“…We deal first with the case of an irreducible space curve in C 3 . Let us see that the problem of determining the limit of q(x, y, z) along X, as well as its computation can be reduced to the case of a real plane curve, a question already addressed in [4]. By Theorem 6, there is a plane curve Y which is birationally equivalent to X, and therefore a local isomorphism µ : X 0 → Y 0 , where X 0 and Y 0 are as in Theorem 6.…”

“…, h m (Y, Z)), and by X ′ i = V (I) ⊂ C 2 the affine variety defined by I, then the limit of q(X, Y, Z) as (X, Y, Z) → O along X i is the same as the limit of q(0, Y, Z) as (Y, Z) → (0, 0) along X ′ i . But the existence of this limit, as well as its value, can be computed using the algorithmic method developed in [4]. By Proposition 4 and Corollary 5 we know that if x is transcendental over C there exists λ ∈ C(x) such that C(x, y, z) = C(x, u), where u = y + λz.…”

“…In [4] Vélez, Cadavid and Molina developed a method for analyzing the existence of limits lim (x,y)→(a,b) q(x, y), where q(x, y) is a quotient of two real analytic functions f and g, under the hypothesis that (a, b) is an isolated zero of g. In the case where f and g are polynomial functions with rational coefficients, the techniques developed in that article provide an algorithm for the computation of such limits, now available in Maple as the limit/multi command [17].…”

“…Our main result is summarized in Theorem 19.In Section 4 we describe an effective method for computing such limits. We provide a high level description of an algorithm that generalizes the one developed in [4], now available in Maple as the limit/multi command.…”