TITLE: Different sizes of infinity
QUESTION [50 upvotes]: Correct me if I'm wrong, but this is what they taught us in precal:
$$\lim_{x\rightarrow\infty}x=\infty$$
$$\lim_{x\rightarrow\infty}x^{2}=\infty$$
But, we also know that $n^{2}>n$ if $n\notin [0,1]$
Does that mean that some infinities are greater than others? Why don't we explicitly define infinity so that we can show differences in sizes?
Thank you in advance!

REPLY [15 votes]: If that seems puzzling to you, this will surely blow your mind:
$$\lim_{x \to \infty} \frac{1}{x} = \lim_{x \to \infty} \frac{1}{x^2} = 0$$
even though $\dfrac{1}{x} > \dfrac{1}{x^2}$ for all $x > 1$.  Are there different sizes of $0$, too?

Actually, there's nothing special about $\infty$ or $0$ in this respect.  It's perfectly possible for two functions to approach the same limit (any limit!) from the same direction, but for one function to be consistently closer to the limit than the other.  This doesn't make the limits any different — it just means that one of the functions approaches the limit faster than the other.
(Of course, $\infty$ as a limit is somewhat special in other ways: when we write $\lim_{x \to a} f(x) = \infty$, what we really mean is that $f(x)$ diverges without bound as $x$ approaches $a$ — or as $x$ grows without bound, if $a$ itself is $\infty$.  But all these special definitions are really only necessary because the real number line with the usual metric we use on it is not topologically compact, and does not even include $\pm \infty$ as proper points.  Instead, we can easily define a metric on the extended real number line, e.g. by homeomorphically mapping it to a closed interval, such that limits at (and to) infinity work just like normal limits.)