What exactly is infinity? On Wolfram|Alpha, I was bored and asked for $\frac{\infty}{\infty}$ and the result was (indeterminate). Another two that give the same result are $\infty ^ 0$ and $\infty - \infty$.
From what I know, given $x$ being any number, excluding $0$, $\frac{x}{x} = 1$ is true.
So just what, exactly, is $\infty$?

First of all, infinity is not a number, it's more like a concept. That's why you can never put infinity on any side of an equation or put it in any mathematical operation. 
For example the following expression is not defined: $ n = \infty$. It's like writing something like $n=apple$. 
It has no mathematical significance, and that's why also any of the following expression aren't defined: $\infty^0$, $ \infty \over \infty$ and so on. 

A lot of people also think that $1/0 = \infty$ but guess what? that is also wrong.
Why? apart from the reasons I just described , notice that $2/0 = \infty$ and so , one can always say: $2/0 = 1/0$ leading to $2=1$ which is absurd. 
On a final note, a mathematical expression can only tend to infinity. When we say that a mathematical expression tends to infinity, we usually mean that for every number we choose, this mathematical expression can be evaluated to be greater.
