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Does the following equality generally hold?

$$ \lim_{x\to\infty, y\to\infty} f(x, y) = \lim_{z\to\infty} f(z, z) $$

If not, what are the necessary conditions for the above equation to hold?

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    It certainly doesn't: consider $$f(x,y) := \tfrac{x}y $$here, $$\lim_{x\to\infty}\lim_{y\to\infty} f(x,y)=\lim_{x\to\infty}0=0$$, but $$\lim_{z\to\infty}f(z,z) = 1.$$ But I don't know what condition would be necessary for the equation to hold.2011-05-30
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    Thanks guys. I got the point. But actually my question is a bit different. I changed the question accordingly. In fact, I want to find the value of $f(.)$ for very large $x$ and $y$.2011-05-30

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