I was asked to give a precise defintion

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