Question about interpreting limit notation: $\lim_{\|x\| \rightarrow \infty}f(x)$

Question

I have a simple question about notation regarding limits, specifically, $$\lim_{\|x\| \rightarrow \infty}f(x).$$

Question:

$\lim_{\|x\| \rightarrow \infty}f(x)$:

In words what we are doing is taking the limit as the "norm" of the point $x$ goes to infinity.

My problem here is I want to make sure I am interpreting the idea behind it correctly. So with taking limits one will encounter an expression of the form: $$\lim_{x \rightarrow \infty}f(x).$$ Here I would visualize our value of $x$ just tending towards "infinity" on a graph. But I'm having trouble visualizing the behaviour in this form $$\lim_{\|x\| \rightarrow \infty}f(x).$$

What it says to me is that the "distance" of the $x$ value is going to infinity. So would a way to visualize it be if we had a fixed point $x_0$ on the number line and we kept on measuring the distance from this fixed point $x_0$ to some arbitrary point $x$ that is really far away (infinity away) from $x_0$? And if this is a valid way of thinking about it, what is the benefit of writing it in this form versus the other way mentioned?

Answer 1

You use the analogy of $\lim_{x \rightarrow \infty} f(x)$, saying

I would visualize our value of just tending towards "infinity" on a graph.

This says to me that you are thinking of functions with domain $\mathbb{R}$. I will also assume that by $||x||$ you mean the Euclidean norm.

In this setting, I think a better analogy would be $$\lim_{x \rightarrow 0 } f(x).$$ Here, you must imagine $x$ approaching 0 from both the left and the right (and observe that it approaches the same value from either side). Similarly, $\lim_{||x|| \rightarrow \infty} f(x)$ is can be visualized as $x$ tending towards both infinity and negative infinity on a graph (with the notation being meaningful if and only if the graph approaches the same limit on either side).

Indeed, this idea generalizes to functions with domain $\mathbb{R}^n$ for any $n \in \mathbb{N}$. Just as $\lim_{x \rightarrow 0} f(x)$ involves $x$ approaching the origin from every possible direction, $\lim_{||x||\rightarrow \infty} f(x)$ involves $x$ going arbitrarily far away from the origin in every possible direction.

Answer 2

It is just notation. We say $\lim_{\|x\| \to \infty} f(x) = L$ iff for any $\epsilon>0$ there is some $B$ such that if $\|x\| > B$ then $|f(x)-L| < \epsilon$.