Rigorous definition of $\lim\limits_{{\mathbf{|x|}\to\infty}}\mathbf{f(x)}=\infty$ $\mathbf{f}:M\to\mathbb{R^m}$ I was asked to give a precise defintion

$\lim\limits_{{\mathbf{|x|}\to\infty}}\mathbf{f(x)}=\infty$ with $\mathbf{f}:M\to\mathbb{R^m}$ and $M\subseteq\mathbb{R^n}$

I know for $$\lim\limits_{{x\to\infty}}f(x)=\infty$$ we would have: $\forall\epsilon >0,\exists$ $\delta >0$ such that $f(x)>\epsilon$ $\forall $ |x|>$\delta$.
I already know the precise definition of the expression $\displaystyle \lim_{x\to\infty}f(x)=\infty$ where $f$ is a scalar function on $\mathbf{R}$. But I want to know how this definition can be generalized to $\displaystyle \lim_{|x|\to\infty}f(x)=\infty$ where $f:M\to\mathbf{R}^m$ is a vector-valued function and $M$ is a subset of $\mathbf{R}^n$.
Also is there a difference between $f(\mathbf{x})$ and $\mathbf{f(x)}$, because my teacher often switches between these two.
Is there anyone who could help me out? I would be very grateful.
 A: $$
\lim\limits_{{\mathbf{|x|}\to\infty}}\mathbf{f(x)}\to\infty
$$
means
For every $A  > 0$, there exists $B > 0$ so that:
for all $\mathbf{x} \in \mathbb R^n$, if
$|\mathbf{x}| > B$, then $|\mathbf{f}(\mathbf{x})|>A$.
I'm guessing $\mathbf{f} : \mathbb R^n \to \mathbb R^m$ and
"$\infty$" on the right means the
extra point for the one-point compactification of $\mathbb R^m$.
A: Let $f:M\to\mathbf{R}^m$ be a function from a subset $M$ of $\mathbf{R}^n$ to $\mathbf{R}^m$.
When $M$ is a bounded subset of $\mathbf{R}^n$, the expression $\displaystyle \lim_{|x|\to\infty}f(x)$ does not make sense because one cannot have "$|x|\to\infty$" for $x\in M$.
Simply assume that $M=\mathbf{R}^n$. Then one can define $\displaystyle \lim_{|x|\to\infty}f(x)=\infty$ as follows:
Definition 1. For every $M>0$, there exists $C>0$ such that $|f(x)|>M$ whenever $|x|>C$. Here $|\cdot|$ denotes the standard Euclidean norm in the corresponding space.

When $m>1$, people may use the notation $\mathbf{f}$ to emphasize that $\mathbf{f}$ is a vector-valued function where the codomain $\mathbf{R}^m$ is of more than dimension one.
Similarly, when $n>1$, $\mathbf{x}$ is used to emphasize the dimension of the domain.
If both $m,n>1$, the boldface $\mathbf{f(x)}$ is used rather than $f(x)$.

One can define anything "rigorously" as one wishes. Definitions themselves are not interesting/useful. What matters is what the definition allows one to do.
One way one may apply Definition 1, is to unify some corner cases of a statement regarding limits.
