Putnam 1999 B3

Let $A=\{(x,y):0\leq x,y<1\}$. For $(x,y)\in A$, let $$S(x,y) = \sum_{\frac{1}{2}\leq \frac{m}{n}\leq 2} x^m y^n,$$ where the sum ranges over all pairs $(m,n)$ of positive integers satisfying the indicated inequalities. Evaluate $$\lim_{(x,y)\rightarrow (1,1), (x,y)\in A} (1-xy^2)(1-x^2y)S(x,y).$$