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

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?

• The second one is incorrect. The left hand side of the symbol $\to$ has to be a function or sequence or something else with a free variable that can be made to tend towards something. Here, the left hand side is just a constant. However, you could have something like $$\left(\lim_{n\to\infty}\frac 1 {m^n}\right)\xrightarrow[m\to\infty]{} 0$$ – Jack M Oct 26 '16 at 17:10
• The nomenclature of the term "limit" implies the arrow you're drawing there. As $x \rightarrow \infty$, $f(x) \rightarrow 0$, but the limit means "what is $f(x)$ limited by as $x \rightarrow \infty$?" There are rigorous mathematical definitions of a limit, and that is not one of them, but the semantics accurately imply the notation in a way that's easy to remember. – Devsman Oct 26 '16 at 17:29
• The limit can't "go" anywhere. It is a fixed number (or infinity) – Jacob Wakem Nov 3 '16 at 0:13

## 3 Answers

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.)

• The limit is not "equal to" infinity. In the common formalism, infinity is not a number. So increasing or decreasing without bound means the limit does not exist, but we use a shorthand of saying the limit is infinity to indicate whether it increases or decreases. Given that, how is it incorrect to say that it tends to infinity? That sounds more correct to me, since it can't be equal. Even if you're correct about the notation, I believe your explanation has severe problems. – jpmc26 Oct 26 '16 at 23:23
• jpmc26, f(x) "tends to" infinity, but lim f(x) is equal to infinity. – Mike Benfield Oct 27 '16 at 3:47
• @jpmc26 I agree with MikeBenfield. I will add that even if it is not rigorous, it's a nice way to see in your mind $f(x)$ tending to something, and $\lim f(x)$ being equal to something. – E. Joseph Oct 27 '16 at 7:48
• @jpmc26 It is not entirely unrigorous to write $\lim f(x)=\infty$, if you really don't want to think of it as a mnemonic then think of it as an element of an extended real number system (by two points compactification) which is completely rigorous. – BigbearZzz Oct 31 '16 at 22:26
• By the way, I am pretty sure that you must have seen the big O notation. That is on the whole different level of potentially misleading notation but I rarely see any one object to that. On the other hand, saying that limit is equal to infinity doesn't lead to any real problem at all. A notations that leads to correct result even when it is "misused" reflect the fact that the intuition behind it is very strong. Discouraging people from saying "the limit is infinity" does more harm than good in my opinion. – BigbearZzz Nov 1 '16 at 9:10

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.

• "Limits equal something." Limits often don't exist. In fact, increasing or decreasing without bound is considered an example of a limit not existing, since infinity is not a number (in the common formalism). So not all limits "equal something." – jpmc26 Oct 26 '16 at 23:22
• @jpmc26 you confuse "the limit exists" and "the limit exists and is finite". It is useful to distinguish no limiting behavior (e.g. oscillation) from unbounded growing in a given direction ($+\infty$, $-\infty$, or even some directed infinity). – Ruslan Oct 27 '16 at 7:42

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