Limit Notation: $ \lim_{x \to \infty} f(x) =\infty $ or $ \lim_{x \to \infty} f(x) \rightarrow \infty$?

Question

If $f(x) \to \infty $ as $x \to \infty $,

Which of the following statements would be more correct:

$I.$ $$ \lim_{x \to \infty} f(x) =\infty $$

Or

$II. $ $$ \lim_{x \to \infty} f(x) \rightarrow \infty $$

I thought $II$ would be more appropriate?

Answer 1

The first one is correct, the second one is incorrect.

Answer 2

You have two usual notations :

$$(1)\qquad \lim_{x\to+\infty}f(x)=+\infty$$

and

$$(2)\qquad f(x)\xrightarrow[x\to+\infty]{}+\infty.$$

So in your case, the first one is correct.

(The second one is not appropriate because $\to$ usually means tends to, and a limit is equal to something.)

Answer 3

First one is correct; "→" generally means "tend(s) to". Limits either equal something (or +/- inf) or don't exist; they do not tend to something.