A question about indeterminate forms

Question

Are there any set of numbers into which any of the indeterminate forms we see in a calculus course, like 0⁰, n/0, 1^infinity, etc has an answer?

I'm asking that because, thanks to the Net, I took notice of other kinds of numbers besides those commonly seen in the high school and most of the university courses: Real and Complex numbers.

There are Hyperreals, Surreals, Quaternions, ... so I thought that some indeterminate form could be have an answer among them, just like we have an answer for the x-th root, x being par, of negative numbers in the set of complex numbers.

Answer 1

$0^0$ is an interesting example.

Since

$$\lim_{x->0} x^x = 1$$

it makes sense to define

$0^0 := 1$

This also has advantages concerning the binomial theorem.

But some mathematicians set $0^0 := 0$ and some say it is indetermined. There seems to be no consense what $0^0$ should be.

The other examples you mentioned are even more problematic.

Answer 2

$0^0$ and $1^\infty$ are indeterminant forms because when you have limits where the pieces approach those parts, any value is possible. They aren't usually defined even in other number systems because it doesn't mesh well with the limits (although some advanced real analysis books will define $0*\infty$ to be $0$, they just do it so they don't have to write separate cases for some formulas).

However, $n/0$ for nonzero $n$ is not an indeterminate form; the only things limits that look like that can do is approach $\infty$ or $-\infty$ (or both due to oscillation, but the point is that the absolute value approaches $\infty$ in any case). As such, there is a number system where that makes total sense: the real projective line, which adds a single "unsigned infinity" so that things which could only be $\pm\infty$ can be defined.