I've seen that wolfram alpha says:
$$\frac{1}{\infty} = 0$$
Well, I'm sure that:
$$\lim_{x\to \infty}\frac{1}{x} = 0$$
But does $\frac{1}{\infty}$ only make sense when we calculate it's limit? Because for me, $1$ divided by any large amount of number will be always almost zero.
On
Yes and no. Think about the implications. Is $0\cdot\infty=1$ ? Because normally, if $\displaystyle\frac ab=c$ , then $a=bc$ But this isn't really the case here, is it ? Because, since all limits of the form $\displaystyle{\lim_{n\to\infty}\frac kn}$ are $0$, for all finite numbers k, then the product $0\cdot\infty$ becomes meaningless. Sometimes it can even be infinity itself, since $\displaystyle{\lim_{n\to\infty}\frac n{n^2}=0}$, for instance. So it is best to avoid such expressions, especially if you are a beginner. Otherwise you'll soon be asking questions like why $\displaystyle\lim_{n\to\infty}\left(1+\frac1n\right)^n=e$ instead of $1$, since, by all appearances, $\frac1\infty=0$, and $\displaystyle\lim_{n\to\infty}1^n=1$.
On
Let's say that you want to divide $1$ pie among an $\infty$ of your friends. Each friend will get $\frac{1}{\infty}$ of the pie or, as you say $0$. After you give out the pie, you rethink matters and say to your friends to give the pie back to you. Each friend will give you his portion of the pie that is a $0$. So you end up with $0$ pies. Where has the pie gone?
A rather interesting argument against $0\cdot\infty =1$ occurred to me. Suppose $0\cdot\infty =1$ holds. Then $1 = \infty \cdot 0 = \infty \cdot (0 \cdot 0) = (\infty \cdot 0) \cdot 0 = 1 \cdot 0 = 0$. Strange, isn't it?
The notation $\displaystyle\frac{1}{\infty}=0$ is used as a shorthand for
The notation $\dfrac 1\infty$ does NOT literally mean "divide $1$ by $\infty$".
So literally, it is nonsense; taken as shorthand for the above, you'll see that notation used pretty commonly when folks evaluate limits. It's what we call "an abuse of notation."