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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. –  leftaroundabout May 30 '11 at 7:55
    
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 $y$. –  Mohsen May 30 '11 at 8:02
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1 Answer

Consider $f(x,y)=e^{-(x-y)^2}$. Then each of the iterated limits is $0$ at infty, and obviously $f(z,z)=1$ for all $z$.

To make the equality hold, you need to change $\lim_{x \to \infty}\lim_{y \to \infty} $ to $\lim_{x,y \to \infty}$. Then, if $f$ has a limit at $\infty$ the equality $\lim_{(x,y) \to \infty}f(x,y)=\lim_{z \to \infty}f(z,z)$ holds.

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Read my comment under the question. Is that what you mean? –  Mohsen May 30 '11 at 8:06
    
If you want to find the approximate value of $f$ for very large $x,y$ then you could calculate $\lim_{z \to \infty}f(z,z)$, but unless you have some particular function in mind, this limit cannot give you a general answer. Maybe you should make some more explanations of what you specifically want in your question. –  Beni Bogosel May 30 '11 at 8:18
    
Ok, here is more explanation: I have a concave function $g(x, y; a, b)$ and I want to show that it's maximum happens at the point $<x^*, y^*>$ for large $a$ and $b$'s. $x$ and $y$ are function variables while $a$ and $b$ are constants. Is it enough to show $\frac{\partial g}{\partial x} = 0$ and $\frac{\partial g}{\partial y} = 0$ for large $a$ and $b$? This is actually what I want to show. –  Mohsen May 30 '11 at 8:42
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