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The product of 0 and anything is $0$, and seems like it would be reasonable to assume that $0 It is also an indefinite form because $$\infty^0 = \exp (0\log \infty) $$ but $\log\infty=\infty$, so the. I'm perplexed as to why i have to account for this condition in my factorial function (trying.

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Is a constant raised to the power of infinity indeterminate It says infinity to the zeroth power Say, for instance, is $0^\\infty$ indeterminate

Or is it only 1 raised to the infinity that is?

It is possible to interpret such expressions in many ways that can make sense The question is, what properties do we want such an interpretation to have $0^i = 0$ is a good. In the context of limits, $0/0$ is an indeterminate form (limit could be anything) while $1/0$ is not (limit either doesn't exist or is $\pm\infty$)

This is a pretty reasonable way to. I began by assuming that $\dfrac00$ does equal $1$ and then was eventually able to deduce that, based upon my assumption (which as we know was false) $0=1$ Why is any number (other than zero) to the power of zero equal to one Please include in your answer an explanation of why $0^0$ should be undefined.

Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number

It seems as though formerly $0$ was. I heartily disagree with your first sentence There's the binomial theorem (which you find too weak), and there's power series and polynomials (see also gadi's answer). Your title says something else than infinity times zero