Limits notation: equals or arrow Recently I was using the following notation to express the limit in a publication:
$$ \lim_{x \rightarrow \infty} f(x) = 0 $$
The reviewer said this is wrong. Instead it should read:
$$ \lim_{x \rightarrow \infty} f(x) \rightarrow 0 $$
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?
 A: 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.
A: 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.
A: 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
$$\lim_{y\to b}\lim_{x\to y} f(x,y)=c.$$
A: When a limit yields +∞ or -∞, it doesn't mean that, the limiting value of that limit is infinity. It actually means that the limit fails to exist by approaching infinity. 
Most importantly +∞ or -∞ are not Real Numbers, so it is so usual that, algebraic or arithmetical rules wouldn't be applied for them.
As for an example:  (+∞)+(-∞)≠0 but may be many of us predict that  (+∞)+(-∞)=0 , which is wrong. 
Sometimes notations are abused or the text books say in a manner, for making us understand but it's not technically and theoritically correct. Because there is no other specific notations for describing that a limit does not exist. So the authors use these notations for making us understand but usually we misunderstand the actual definition or meaning lying behind of these notations. 
In the context of the real line, say for Calculus, ∞ is not a real number, so it's not proper to say that anything "equals" ∞. In particular, a limit can't equal this. 
May be that's why your supervisor told you to use this of arrow notation. I think it is fully correct but due to lack of knowing the limits behavior this symbol seems quite illegal or mathematically error. 
