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.

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In this paper we study initial topological properties of the (non-)finitely-generated locus of Frobenius Algebra coming from Stanley-Reisner rings defined through face ideals. More specifically, we will give a partial answer to a conjecture made by M. Katzman about the openness of the finitely generated locus of such Frobenius algebras. This conjecture can be formulated in a precise manner: Let us definewhere F denotes the Frobenius functor and E Rp the injective hull of the residual field of the local ring (Rp, pRp). Is the locus U an open set in the Zariski Topology? In the case where R is a ring of the form R = K[[x 1 , . . . , xn]]/I, where I ⊂ R is a face ideal, i.e., square-free monomial ideal, we show that the corresponding U has non-empty interior. Even more, we prove that in general U contains two different types of opens sets and that in specific situation its complement contains intersections of opens and closes sets in the Zariski topology. Furthermore, we explicitly verify in some non-trivial examples that U is an non-trivial open set.