What is infinity times the reciprocal of infinity?

Question

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 $1$?

Answer 1

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.