Comic Strips

the concept of infinity + 1 can be rigorously defined (as an ordinal number). the basic idea is that infinity +1 is the set containing every single positive whole number, in increasing order, and then something else.

but what you said about infinity in calculus is correct. the “infinity” that appears in calculus is conceptually a different idea of infinity and it’s basically just an inconvenient choice of notation that they’re called the same thing.

in complex analysis, there’s also the riemann sphere, which is basically a way to view the sphere as the complex plane in addition to the “point at infinity”. i.e., 0 is the south pole, and infinity is the north pole. and in this context it’s fairly common to say stuff like “f(infinity) = 0” or “f(2) = infinity”. these can all be understood in terms of limits as you described, but it does sort of blur the line between “actual value” and “potential value”, since infinity is actually a point in the riemann sphere, but it’s primarily described in terms of limits.