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This is a question of notation. I have seen in many articles that people often denote $+\infty$ when talking about 'positive infinity' of the real numbers. Is that a convention, or it can be written as anyone pleases? I never liked the notation $+\infty$ because it seemed that the $+$ sign is redundant. In my opinion there is no confusion if someone writes $\infty$ for the positive infinity and $-\infty$ when talking about the negative infinity.

Still, the fact that I've seen the $+\infty$ notation in almost every article I've read in a while made me ask this question.

Is the $+$ in the notation $+\infty$ necessary? Do $\infty$ and $+\infty$ mean the same thing? (of course I'm talking about the real line here)

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The "+" is optional and used for emphasis, in my experience. I liken it to how come composers put a "natural sign" in a measure to emphasize that there is no longer an accidental (sharp or flat) being applied to a note. –  The Chaz 2.0 Apr 3 '12 at 14:44
Confusion can arise with the unsigned infinity of the projective line, which is also denoted $\infty$. For example, $\lim_{x \to 0} 1/x = \infty$ in the projective line, but $\lim_{x \to 0} 1/x \neq +\infty$ in the extended reals. –  Chris Eagle Apr 3 '12 at 15:04
I can't resist putting in my $+2$ cents' worth. –  André Nicolas Apr 3 '12 at 15:17
@AndréNicolas: It took me $+$ a minute to catch it... –  The Chaz 2.0 Apr 3 '12 at 15:43
I once had a reviewer complain about writing $h: (0, \infty) \to (\frac{1}{2}, \infty)$ instead of $h: (0, +\infty) \to (\frac{1}{2}, +\infty)$. It seemed to me that in the context of the story, the $+$ was unnecessary, so I did not change it. It's a matter of taste, I guess. –  TMM Apr 3 '12 at 17:33
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2 Answers

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The answer to your question depends on individual opinion/definition. So here is my opinion.

I take $\infty$ to mean $+\infty$. Why? Because if you insist that one has to write plus in front $\infty$ every time one means positive infinity, then it is like saying that the symbol $\infty$ isn't well defined. So why not just adopt the convention from the real numbers where $+x$ means $x$. We don't write $+1$, we just write $1$.

Now that said, if you are writing a paper where it is essential that the reader catches whether something is $\infty$ or $-\infty$, then you might want to add the plus-sign in front when you mean (positive) infinity.

Or, if a limit is equal to either positive or negative infinity you might write $\pm \infty$ (thereby indirectly writing a $+$.

That is my opinion.

Note for example that in Stewart's calculus book the interval from negative infinity to (positive) infinity is written $(-\infty , \infty)$, so different from what Thomas Andrews has come across in his answer.

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As with all notation, it depends on the context. For example, when dealing with a sequence $a_1,...,a_n,...$ we write $\lim_{n\to\infty} a_n$. On the other hand, the value of this limit might be $+\infty$ or $-\infty$. So you can sometimes write:

$$\lim_{n\to\infty} a_n = -\infty$$

On the other hand, when dealing with a function on the real line, say, $f(x)=\frac{e^x}{1+e^x}$, the behavior exists and is different for large negative and large positive numbers. So we dinstinguish:

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


$$\lim_{x\to -\infty} f(x)=0$$

In this case, it doesn't make sense to talk about $\lim_{x\to\infty} f(x)$.

Other places you'll see infinite values are in intervals, like:

$$[a,+\infty)$$ $$(-\infty,b]$$ $$(-\infty,+\infty)$$

The key is to realize that $\infty$ in all these instances are shorthands for definitions. So $[a,+\infty)$ is the set of all real numbers at least as big as $a$, for example.

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