Does $\lim\limits_{x\to\infty}$ equal to $\lim\limits_{x\to+\infty}$?

Question

As per title, does $$\lim\limits_{x\to\infty}$$

mean $\lim\limits_{x\to+\infty}$ or $\lim\limits_{x\to\pm\infty}$?

This link seems to tell me that it's the latter: https://qc.edu.hk/math/Certificate%20Level/Limit%20mistake.htm

However, evaluating the limit in WolframAlpha however, gave me a different answer: https://www.wolframalpha.com/input/?i=limit+of+x(sqrt(x%5E2%2B1)-x)+as+x+approaches+infinity

So WolframAlpha seems to consider the former identity.

Any clarifications would be welcomed.

score 3 · Accepted Answer · answered Dec 13 '18 at 08:09

3

When we write

$$\lim_{x\to\infty}$$

we usually mean

$$\lim_{x\to+\infty}$$

when we need to be more clear in that the latter is preferable.

When we want indicate the two possiblities we can use

$$\lim_{|x|\to\infty}$$

answered Dec 13 '18 at 08:09

user

154,566

score 1 · Answer 2 · answered Dec 13 '18 at 08:11

It means positive infinity.

It's like how $2$ implicitly means $+2$ or $60$ means $+60$. Similarly, $\infty$ means $+\infty$: unless a number is negative, in which case the negative sign is applied, it is implicitly positive. (Unless it's $0$ which is neither positive or negative. So you could say that having no sign means the number is "nonnegative" instead.)

If we want to speak of a limit approaching negative infinity, it'll be denoted $x \to -\infty$. If we speak of a limit approaching either infinity, we say the absolute value of the variable approaches infinity, i.e. $|x| \to \infty$. (Notice how $x \to -\infty$ and $x \to \infty$ both imply $|x| \to \infty$.)

score 1 · Answer 3 · answered Dec 13 '18 at 08:18

1

Generally speaking + $\infty$ is same as $\infty$ while -$\infty$ is approaching a point at infinity in opposite direction. For e.g $e^x$ has different limits at + $\infty$ and - $\infty$.

I hope this helps.

answered Dec 13 '18 at 08:18

Devendra Singh Rana

1,780
1
15
32

score 1 · Answer 4 · answered Dec 13 '18 at 08:21

This is simply a matter of notation, which is by no means universal.

The link you've given shows one example where they mean either. Wolfram Alpha seems to automatically only consider the limit at $+\infty$.

This is not universal, and I've seen both commonly enough. Therefore, it seems important to read from the actual context what they mean, when they use the notation. (Do they define the limit at infinity?)

My personal opinion is that I think the notation $\lim\limits_{x\to\infty}$ should only be used to denote $\lim\limits_{x\to+\infty}$ if $x\in\mathbb R$. But unfortunately, it seems like my opinion is not universal. Therefore, you have to make sure in every context.

Does $\lim\limits_{x\to\infty}$ equal to $\lim\limits_{x\to+\infty}$?

4 Answers4