TITLE: There is no smallest infinity in calculus?
QUESTION [20 upvotes]: Somewhat of a basic question, but I tried mixing set theory and calculus and the result is a giant mess.
From set theory (assume ZFC) we know there is a smallest infinite cardinal, $\aleph_0$, and that infinite numbers are well ordered, $\aleph_1 > \aleph_0$ etc
Now if we move to the world of calculus, even there, there is a difference between one infinity and the other.
$\lim_{x \to \infty} x = \infty$, and $\lim_{x \to \infty} e^x = \infty$, but they are not the same, you could say that the $e^x$ one is bigger, because $\lim_{x\to \infty} \frac{e^x}{x} = \infty$ as can be shown easily with L'Hôpital's rule.
This leads me to believe that unlike in set theory, in calculus there is no smallest infinity, since if $\lim_{x \to \infty} f(x)= \infty$, then $\lim_{x \to \infty} \log (f(x) = \infty$ but a smaller $\infty$.
So which version is "correct"? Is there really a smallest infinity like in set theory? or we can keep getting smaller and smaller to no end like in calculus? Or both are correct in different context? I'm a bit confused.
Which also leads to another question, when we say in calculus that some limit tends to $\infty$, which $\infty$ are we talking about? $\aleph_2$? $\aleph_0$?

REPLY [2 votes]: Let's nail one thing straight away. $+\infty$ is not a number.
$$\lim_{x \to a} f(x) = +\infty$$
is not an equality. It is a notational convention that describes the growth without bound of the function at the point $x=a$.