Real valuations and the limits of multivariate rational functions

The purpose of this paper is to investigate the limits of multivariate rational functions with the aid of the theory of real valuations. The following is one of our main results. For two nonzero polynomials f, g ∈ ℝ[x1,…,xn] and (a1,…,an) ∈ ℝn, the (finite) limit of the rational function [Formula: see text] at (a1,…,an) does not exist if and only if (1) there exists a sequence u1(x),…,un(x) of polynomials over ℝ in one variable x such that ui(0) = ai for i = 1,…,n, g(u1(x),…,un(x)) ≠ 0, but [Formula: see text]; or (2) there exist two sequences u1(x),…,un(x) and w1(x),…,wn(x) of polynomials over ℝ in one variable x such that ui(0) = wi(0) = ai for i = 1,…,n, g(u1(x),…,un(x)) ⋅ g(w1(x),…,wn(x)) ≠ 0, but [Formula: see text].