# Limits notation: equals or arrow

Recently I was using the following notation to express the limit in a publication:

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

$$\lim_{x \rightarrow \infty} f(x) \rightarrow 0$$

Is there a semantic difference between these two expressions? I did not find anything that would clarify the difference. In case it is just a matter of notational preference: Would you agree that the former notation is more common and maybe "more correct" since the limit actually is equal to the right hand side?

• I think the 2nd one is just plain wrong, it should be $f(x)\rightarrow 0$ as $x\rightarrow\infty$ may i ask which journal is it? – Lost1 Apr 11 '13 at 11:37
• The reviewer is flat-out wrong, and his notation is incorrect. Either your version or $f(x)\to 0\text{ as }x\to\infty$ (or the like) is acceptable, but not his strange hybrid. – Brian M. Scott Apr 11 '13 at 11:37
• I have never seen the second notation. Your notation is the way to go. – ronash Apr 11 '13 at 11:38
• If you have a proof that this referee told you this, you should contact the editors to let them know. Hopefully he/she will never have a referee job in this journal again. – Julien Apr 11 '13 at 12:05
• Thank you very much for confirming that sometimes I'm not completely wrong. It was a paper in the field of data mining, and proper notation is sometimes slightly left behind here. But needless to say that this won't make the reject (and reading of reject justifications) more enjoyable... – bluenote10 Apr 11 '13 at 12:31

I think the first one is right and the second is wrong. When the limit existis it is surely a number, and a number doesn't tend to anything.

Maybe he meant something like $$f(x)\overset{x\to \infty}{\to} 0$$ But I would prefer always $$\lim_{x\to \infty} f(x)=0$$ because the limit IS something and not tends to something.

When you use the arrows you say something like it tends to so essentially you say in $f(x)\to 0$ as $x\to \infty$ that when $x$ goes to $\infty$ your function tends to zero.

The second notation would be, as the limit is a fixed $c$, $$\text{c}\to 0$$ which I think is nonsense.

The notation to use depends on wheter you are in a text or you are in display mode. In a display mode I would use $$\lim_{x\to \infty} f(x)=0$$ In inline there are three options, the first is

• $\lim_{x\to \infty} f(x)=0$
• $f(x)\to 0$ as $x\to \infty$
• As $x$ goes to infinity $f(x)$ tends to zero.

Personally I prefer the third, because in the first the index will be hardly legible, in the second there are to many mathematical symbols in a sentence and the third will be the easiest to read.

There are two way to express it:

1) $\lim_{x \rightarrow \infty} f(x) = 0$

2) $f(x) \rightarrow 0$, as $x \rightarrow \infty$.

I believe they are same, no difference.

• The reviewer doesn't use any of those. How does it answer the question ? – Dominic Michaelis Apr 11 '13 at 11:45
• I think he is wrong. You may tell him the fact! – Paul Apr 11 '13 at 11:47

Of course a hybrid notation such as $$\lim_{x\to a} f(x,y)\to c\qquad\text{as }y\to b$$ might be possible, which would be just the same as $$\lim_{y\to b}\lim_{x\to y} f(x,y)=c.$$

• Urgh. ${}{}{}{}$ – vonbrand Apr 11 '13 at 12:33