Certain infinities can grow faster than others, though. That’s why L’Hôpital’s rule works.
For example, the area of a square of infinite size will be a “bigger” infinity than the perimeter of an infinite square (which will in turn be a bigger infinity than the infinity that is the side length). “Bigger” in the sense that as the side length of the square approaches infinity, the perimeter scales like 4*x but the area scales like x^2 (which gets larger faster as x approaches infinity).
It might give use different growth rate but Infinity is infinite, it’s like the elementary school playground argument saying “infinity + 1” there is no “infinity + 1”, it’s just infinity. Infinity is the range of all the numbers ever, you can’t increase that set of numbers that is already infinite.
but in this case we are comparing the growth rate of two functions
oh, you mean like taking the ratio of the derivatives of two functions?
it’s like the elementary school playground argument saying “infinity + 1” there is no “infinity + 1”, it’s just infinity
but that’s not the scenario. The question is whether $100x is more valuable than $1x as x goes to infinity. The number of bills is infinite (and you are correct that adding one more bill is still infinity bills), but the value of the money is a larger $infinity if you have $100 bills instead of $1 bills.
Edit: just for clarity, the original comment i replied to said
Lhopital’s rule doesn’t fucking apply when it comes to infinity. Why are so many people in this thread using lhopital’s rule. Yes, it gives us the limit as x approaches infinity but in this case we are comparing the growth rate of two functions that are trying to make infinity go faster, this is not possible. Infinity is infinite, it’s like the elementary school playground argument saying “infinity + 1” there is no “infinity + 1”, it’s just infinity. Infinity is the range of all the numbers ever, you can’t increase that set of numbers that is already infinite.
Those are all aleph 0 infinities. There’s is a mathematical proof that shows the square of infinity is still infinity. The same as “there is the same number of fractions as there is integers” (same size infinities).
Certain infinities can grow faster than others, though. That’s why L’Hôpital’s rule works.
For example, the area of a square of infinite size will be a “bigger” infinity than the perimeter of an infinite square (which will in turn be a bigger infinity than the infinity that is the side length). “Bigger” in the sense that as the side length of the square approaches infinity, the perimeter scales like
4*xbut the area scales likex^2(which gets larger faster asxapproaches infinity).It might give use different growth rate but Infinity is infinite, it’s like the elementary school playground argument saying “infinity + 1” there is no “infinity + 1”, it’s just infinity. Infinity is the range of all the numbers ever, you can’t increase that set of numbers that is already infinite.
oh, you mean like taking the ratio of the derivatives of two functions?
but that’s not the scenario. The question is whether $100x is more valuable than $1x as x goes to infinity. The number of bills is infinite (and you are correct that adding one more bill is still infinity bills), but the value of the money is a larger $infinity if you have $100 bills instead of $1 bills.
Edit: just for clarity, the original comment i replied to said
Those are all aleph 0 infinities. There’s is a mathematical proof that shows the square of infinity is still infinity. The same as “there is the same number of fractions as there is integers” (same size infinities).