1. **Problem:** Determine if $$\lim_{(x,y) \to (1,1)} \frac{xy}{xy}$$ exists. 2. **Formula and rules:** The limit of a function $$f(x,y)$$ as $$(x,y) \to (a,b)$$ exists if the value approaches the same number regardless of the path taken. 3. **Intermediate work:** Simplify the expression: $$\frac{xy}{xy} = 1 \quad \text{for all } (x,y) \neq (0,0)$$ 4. **Explanation:** Since the function simplifies to 1 everywhere except possibly at points where $xy=0$, and the limit point is $(1,1)$ where $xy=1 \neq 0$, the limit is simply 1. 5. **Final answer:** $$\lim_{(x,y) \to (1,1)} \frac{xy}{xy} = 1$$