Meaning of notation $\lim_{|x|+|y|\rightarrow\infty}|f(x,y)|$

Question

I recently stumbled upon the statement "$|f(x,y)|\rightarrow\infty$ as $|x|+|y|\rightarrow\infty$" in Robert Israel's answer at What does it mean for a level curve to be closed or open?.

What does this notation mean? I think it is equivalent to the following condition:

For any curve, not necessarily a level curve, $C:[0,\infty)\rightarrow\mathbb{R}^2$ where $C(t):=(x(t),y(t))$, $\lim_{t\rightarrow\infty}(|x(t)|+|y(t)|) = \infty$.

In general, given $g,h:\mathbb{R}^2\rightarrow\mathbb{R}$, when does the notation $\lim_{g(x,y)\rightarrow\infty}h(x,y)$ make sense?

Answer 1

Formally, $$\lim_{g(x,y) \to \infty} h(x,y) = \infty$$ means

For any real M, there is some N such that [for any $x,y$,] $g(x,y) > N$ implies $h(x,y) > M$.

In the equivalent contrapositive, this is

For any real M, there is some N such that [for any $x,y$,] $h(x,y) \leq M$ implies $g(x,y) \leq N$.

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Using this form, the point of Robert's comment can be made more clear: Consider the level curve $f(x,y) = c$ and suppose the limit in Robert's post is true. By setting $M=|c|+1$, we conclude there is some $N$ such $|f(x,y)| \leq |c|+1$ implies $|x|+|y| \leq N$. In particular, on the whole of the level curve $f(x,y) = c$, we have $|x| + |y| \leq N$, which is bounded (it's a square along with its interior), so the level curve $f(x,y) = c$ is bounded for each $c$.