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Under what circumstances does the existence of the iterated limit $\lim\limits_{x \to \infty} \left(\lim\limits_{y \to \infty}\ a_{x,y} \right)$ imply the existence of the double limit $\lim\limits_{(x,y) \to (\infty,\infty)} a_{x,y}$?

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For sequences, see this previous question. See also the Moore-Osgood Theorem. –  Arturo Magidin Jun 3 '11 at 2:04
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