An expression like 0/0, called an indeterminate form, can be thought of in a different way in terms of infinitesimals. Suppose I tell you I have two infinitesimal numbers and in my pocket, and I ask you whether is finite, infinite, or infinitesimal. You can't tell, because and might not be infinitesimals of the same order of magnitude. For instance, if , then is finite; but if , then is infinite; and if , then d/e is infinitesimal. Acting this out with numbers that are small but not infinitesimal,
On the other hand, suppose I tell you I have an infinitesimal number and a finite number , and I ask you to speculate about . You know for sure that it's going to be infinitesimal. Likewise, you can be sure that is infinite. These aren't indeterminate forms.
We can do something similar with infinite numbers. If and are both infinite, then is indeterminate. It could be infinite, for example, if was positive infinite and . On the other hand, it could be finite if . Acting this out with big but finite numbers,
Example
If is a positive infinite number, is finite, infinite, infinitesimal, or indeterminate?
Trying it with a finite, big number, we have
: H=1/d d^-1 : sqrt(H+1)-sqrt(H-1) d^1/2+0.125d^5/2+...
For convenience, the first line of input defines an infinite number in terms of the calculator’s built-in infinitesimal . The result has only positive powers of , so it’s clearly
infinitesimal.
More rigorously, we can rewrite the expression as . Since the derivative of the square root function evaluated at is , we can approximate this as
which is infinitesimal.
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