Prove that a specific double integral limit results in a non-negative integer | Step-by-Step Solution

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I see you're working with a fascinating complex analysis problem involving Gaussian functions and double integrals! This type of problem typically connects several powerful concepts from complex analysis, including contour integration, residue theory, and properties of analytic functions. Here's what I'd like you to think about first: what specific mathematical expression are you working with, and what parameter is taking a limit? Also, when a problem states that a limit "always results in a non-negative integer," this is often a big clue that you're actually counting something mathematical - perhaps zeros, poles, or some topological property. I'd encourage you to share the complete integral expression so we can explore which tools from complex analysis (like the residue theorem or Cauchy's integral formula) might be most helpful here. Remember, problems that guarantee integer results usually have elegant underlying reasons that become clear once you identify the right approach!