TITLE: $\frac{1}{\infty}$ - is this equal $0$?
QUESTION [5 upvotes]: 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.

REPLY [6 votes]: 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?