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I know that $\infty/\infty$ is not generally defined An example of an infinite number in $ {}^\ast \mathbb r$ is represented by the sequence $1,2,3,\ldots$. However, if we have 2 equal infinities divided by each other, would it be 1

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Infinity refers to something without any limit, and is a concept relevant in a number of fields, predominantly mathematics and physics Thus both the square root of infinity and square of infinity make sense when infinity is interpreted as a hyperreal number The english word infinity derives from latin infinitas, which can be translated as unboundedness , itself derived from the greek word apeiros, meaning endless .

Can this interpretation (subtract one infinity from another infinite quantity, that is twice large as the previous infinity) help us with things like $\lim_ {n\to\infty} (1+x/n)^n,$ or is it just a parlor trick for a much easier kind of limit?

In particular, infinity is the same thing as 1 over 0, so zero times infinity is the same thing as zero over zero, which is an indeterminate form Your title says something else than infinity times zero It says infinity to the zeroth power. Similarly, the reals and the complex numbers each exclude infinity, so arithmetic isn't defined for it

And then, you need to start thinking about arithmetic differently. Infinity plus infinity ask question asked 13 years, 7 months ago modified 6 months ago 7 neither the maximum or supremum of a subset are guaranteed to exist If you consider the real numbers as a subset of itself, there is no supremum

If you consider it a subset of the extended real numbers, which includes infinity, then infinity is the supremum.

Infinity divided by infinity ask question asked 7 years, 10 months ago modified 7 years, 10 months ago The infinity can somehow branch in a peculiar way, but i will not go any deeper here This is just to show that you can consider far more exotic infinities if you want to Let us then turn to the complex plane