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Suppose $f:\mathbb{R}^2\to\mathbb{R}$ and we define things like $f(x,+\infty):=\lim_{y\to+\infty}f(x,y)$. Then how would one define, say, $f(+\infty,-\infty)$?

Typically for limits not at infinity for both functions on $\mathbb{R}$ and $\mathbb{R}^2$ we do something like

$$\lim\limits_{x\to c}g(x)=y\iff\forall\,\varepsilon>0\,\,\exists\,\delta>0 \mbox{ s.t. }x\in B(c,\delta)\implies g(x)\in B(y,\varepsilon).$$

Then this has a natural alteration when one wishes to take $c$ to be $+\infty$ (in the case $x\in\mathbb{R}$). However, I don't see how alter this when $x\in\mathbb{R}^2$.

EDIT: I'd be very appreciative if someone answered my question outright, but if otherwise someone has a source which deals with this I'll be happy too.

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You can certainly define it like you would think: $$\lim\limits_{x\to =\infty, y \to -\infty}f(x,y)=c\iff\forall\,\varepsilon>0\,\,\exists\,m,n>0 \mbox{ s.t. }x \gt m \wedge y \lt -n \implies f(x,y)\in B(c,\varepsilon)$$ which says that if $x$ gets large enough positive and y gets large enough negative, then $f(x,y)$ is close to $c$. As with two dimensional limits not at infinity, you have to have the limit be the same no matter what path you take to get there.

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