# What is infinity times the reciprocal of infinity?

I was talking with a friend about interesting properties of numbers and their theoretical contradictions and solutions when we came up with this. What is the answer?

So...
$x * ∞ = ∞$
and...
$\frac{1}{x}*x=1$

So what do you get when you do...
$\frac{1}{∞}*∞$? $∞$ or $1$?

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But infinity isn't a number... – J. M. Dec 10 '11 at 5:38
@J.M. But it can still be used in equations. – JShoe Dec 10 '11 at 5:42
@JShoe: No, it can't. It isn't well defined. – mixedmath Dec 10 '11 at 5:43
Usually in measure theory, Lebesgue integrals and things like that, we avoid "to divide" by 0 and by $\infty$. To prove things, these cases are considered apart. I'm with J.M. infinite isn't a number. – leo Dec 10 '11 at 5:44

Unfortunately, infinity is not a number, and cannot be manipulated as such.

For example, $\lim_{n \to \infty} x = \infty$, and $\lim_{n \to \infty} x^2 = \infty$, but

$\lim_{n \to \infty} \frac{x}{x} = \frac{\infty}{\infty} = 1$,

$\lim_{n \to \infty} \frac{x}{x^2} = \frac{\infty}{\infty} = 0$,

$\lim_{n \to \infty} \frac{x^2}{x} = \frac{\infty}{\infty} = \infty$,

In fact, we can make $\frac{\infty}{\infty}$ converge to anything we want.

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...Huh? That seems wrong, but hey, everything does until you understand it. Thanks! – JShoe Dec 10 '11 at 5:57
Infinity isn't something that easily succumbs to intuition, @JShoe. :) – J. M. Dec 10 '11 at 6:01
Those $=$ signs above need to be taken very loosely... – LarsH Apr 13 '12 at 16:03