I'm trying to gain a more intuitive understand of the concept of an infinitesimal. When physicists speak of an infinitesimal quantity, if that basically the same as the first order term in a Taylor series expansion of a function they're analyzing?
I think one good way to get an intuitive feel for an infinitesimal is by means of the following challenge: "You give me any small number, as small as you wish, and I'll give you a number even smaller, and yet, an infinitesimal is smaller even than my number."
If we both start this endless game, we might quickly get bored, but hopefully, we would come away with a sense of where we'd go if we went on forever playing.
A similar challenge could be used to get a sense of the infinitely large, or a sense of what it means to approach an asymptotic limit.