I considered deleting the post, but this seems more cowardly than just admitting I was wrong. But TIL something!
this post was submitted on 06 Jan 2024
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for $1 bills: lim(x->inf) 1*x
for $100 bills: lim(x->inf) 100*x
Using L'Hôpital's rule, we take the derivative of each to get their ratio, ie: 100/1, so the $100 bill infinity is bigger (since the value of the money grows faster as the number of bills approaches infinity, or said another way: the ratio of two infinities is the same as the ratio of their rates of change).
The conventional view on infinity would say they're actually the same size of infinity assuming the 1 and the 100 belong to the same set.
You're right that one function grows faster but infinity itself is no different regardless of what you multiply them by. The infinities both have same set size and would encompass the same concept of infinity regardless of what they're multiplied by. The set size of infinity is denoted by the order of aleph (ℵ) it belongs to. If both 1 and 100 are natural numbers then they belong to the set of countable infinity, which is called aleph-zero (ℵ₀). If both 1 and 100 are reals, then the size of their infinities are uncountably infinite, which means they belong to aleph-one (ℵ₁).
That said, you can definitely have different definitions of infinity that are unconventional as long as they fit whatever axioms you come up with. But since most math is grounded in set theory, that's where this particular convention stems from.
Anyways, given your example it would really depend on whether time was a factor. If the question was "would you rather have 1 • x or 100 • x dollars where x approaches infinity every second?" well the answer is obvious, because we're describing something that has a growth rate. If the question was "You have infinity dollars. Do you prefer 1 • ∞ or 100 • ∞?" it really wouldn't matter because you have infinity dollars. They're the same infinity. In other words you could withdraw as much money as you wanted and always have infinity. They are equally as limitless.
Now I can foresee a counter-argument where maybe you meant 1 • ∞ vs 100 • ∞ to mean that you can only withdraw in ones or hundred dollar bills, but that's a synthetic constraint you've put on it from a banking perspective. You've created a new notation and have defined it separately from the conventional meaning of infinity in mathematics. And in reality that is maybe more of a physics question about the amount of dollar bills that can physically exist that is practical, and a philosophical question about the convenience of 1 vs 100 dollar bills, but it has absolutely nothing to do with the size of infinity mathematically. Without an artificial constraint you could just as easily take out your infinite money in denominations of 20, 50, 1000, a million, and still have the same infinite amount of dollars left over.