Limits of quotients of bivariate real analytic functions

Necessary and sufficient conditions for the existence of limits of the form are given, under the hypothesis that f and g are real analytic functions near the point (a,b), and g has an isolated zero at (a,b). The given criterion uses a constructive version of Henselʼs Lemma which could be implemented in a computer algebra system in the case where f and g are polynomials with rational coefficients, or more generally, with coefficients in a real finite extension of the rationals. A high level description of an algorithm for determining the existence of the limit as well as its computation is provided.