TITLE: Which infinity is meant in limits?
QUESTION [42 upvotes]: For example, when we write $\lim_{x\rightarrow \infty} f(x)$ - which infinity is meant and why? Countable? If uncountable - which and why?

REPLY [14 votes]: The infinity here is a point at infinity in the two-point compactification $\mathbb{R} \cup \{ -\infty, +\infty \}$ of the real line. This is just one of many meanings of "infinity" in mathematics; see, for example, this math.SE question.