There is no such thing as ∞, there is only [∞]. x/∞ = NaN. It's exactly as meaningful as x/Tuesday or x/your_face.
To define [∞] you need two things. A specific number in question, and a precision. Let's pick 333. Precision say ±0.01%. [∞] is then any number such that 333/[∞] is 0 to within 0.01%, that is, smaller than 0.00005, meaning for our purposes [∞] is 6,660,100 and all larger numbers. Note the placement of the one is at five significant figures. To be precise [∞] >= 6.6601*10^6.
Let's recurse. We can easily see that [∞] exists for any integer or real number, because division works. That's the [∞] mathematicians like to talk about - the [[∞]], basically.
We can also see [∞] does not exist for [∞]. If we divide [∞] by 5*10^-5, we don't get a number, we get nonsense. [∞] is not an integer or a real number, it is a set of numbers. You can multiply any particular entry by 20,000 if you want, but that doesn't change which members belong in the set.
You can see that no matter how precise you need to be, there will be at least one member for [∞]. The set is never empty. However, statements such as 'infinitely precise' are meaningless. You can't have a number ±[4, 12, 0.09, 333]. At best that's four numbers, not 'a' number. What about [∞] defined on 333 ± 1/[∞]? Recursive, doesn't converge. NaN.
Numbers with finite precision do not approximate nature. Nature is inherently finite in precision, because infinity is not a number, and precision is a number.