Limits notation: equals or arrow

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.

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 $$\lim_{y\to b}\lim_{x\to y} f(x,y)=c.$$