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So, here is a question about two-variable-limit and iterated limits that needs an example.

Giving a doubly indexed array $a_{mn}$ where $m,n \in \Bbb N$.
Then construct an example where lim$_{m,n\rightarrow\infty}$ $a_{mn}$ exists but neither the iterated could be computed.

I have a basic idea if it was asking lim$_{m,n\rightarrow(0,0)}$, but now I'm stuck with this question.

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    How are you defining $\lim_{m,n \to \infty}a_{mn}$? – 2017-02-14
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    That's why even the question itself is saying that $a_{mn}$ is doubly indexed. But of course lim$_{m,n\rightarrow\infty}$ $a_{mn}$ here means lim$_{m\rightarrow\infty,n\rightarrow\infty}$ $a_{mn}$, as both $m,n\rightarrow\infty$ together instead of seperately. – 2017-02-14

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What about $$a_{mn}=\frac{(-1)^m}{n}+\frac{(-1)^n}{m}.$$

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    Uh-huh! That's clever. In this case if both $m,n\rightarrow\infty$ together $a_{mn} \rightarrow 0$ and iterated cannot be computed since $(-1)^m$ or $(-1)^n$ cannot be computed separately. Thanks! – 2017-02-14