Meaning of $\lim_{x\to \infty}f(x) = \infty$

Question

Let $x \in \mathbb{R}^m, f:\mathbb{R}^m \to \mathbb{R}^n$. Explain what means $\lim_{x \to \infty}f(x) = \infty$

Sorry if the question is rather simple, unfortunatelly I couldn't find an answer on my textbooks.

How can I define what means to a vector go to infinity? Does it mean all of it's coordinates go to infinity or that $|x|$ tends to infinity?

for $n=1$ I have that $\lim_{x\to \infty}f(x) = \infty$ means that $\forall M>0$ exists $H>0$ such that $\forall x>H \implies f(x)>M$.

but how can I match this for $x\in \mathbb{R}^n$? If I use the norm I could have each coordinate of x or f(x) going to $-\infty$ and still $|x|>H$ and $|f(x)|>M$

The definition would be with $H \in \mathbb{R}^m$ and $M \in \mathbb{R}^n$ and the inequalities would be $x_i>h_i \forall i=1,...,m \implies f(x)_j>m_j\ \forall j=1,...,n$ ?

Answer 1

Talking about vectors, there's no $\pm\infty$ in general (unless you talk about a certain coordinate), just $\infty$. And the limit is usually defined with the correspondent norm, i. e., $\lim_{x\to\infty}f(x)=\infty$ means that $\forall N>0$ exists $M>0$ such that if $||x||>M$ then $||f(x)||>N$.

Answer 2

When you have $\infty$ in any kind of limit notation on $\mathbb{R}^n$, $n \neq 1$, you must expect (if not told otherwise) to be dealing with the one-point compactification, for which the notation is clear and the limit is the usual topological limit:

$\lim\limits_{x \to \infty^m}f(x)=\infty^n$

(where the $m,n$ are there for the sake of explicity), which means precisely that for every neighbourhood $V$ of $\infty^n$, there exists a neighbourhood $U$ of $\infty^m$ such that $f$ takes $U\backslash \{\infty^m\}$ inside $V$. Since neighbourhoods of $\infty$ are complements of compact sets (by definition), one can show that this is equivalent to saying that for every $M>0$, there exists $K>0$ such that if $\Vert {x}\Vert >K$, then $\Vert f(x)\Vert>M$.

If $n=1$ or $m=1$, the notation can be a bit confusing with the also standard two-point compactification $[-\infty,+\infty]$. Some people prefer to explicitly put a sign in front of $\infty$ (denoting thus by $+\infty,-\infty$) if they are dealing with the two-point compactification, in order to implicitly tell that they are not dealing with the one-point compactification.