It is known that $0^0$ despite being an indeterminate limit form, is usually defined to be equal to $1$. I wonder whether similar conventions exist for some other "indeterminate forms" in the context of two-point compactifications of real numbers. It would be great if someone showed that some authors used these conventions.
Particularly I am interested to know about usage of the following conventions:
$1^\infty=1$
$0 \cdot \infty=0$
$\infty^0=1$
$\frac 00=0$
I also would be interested whether any author proposed distinguishing between "definable" indeterminate forms (those which can be conveniently defined to have certain value, like $0^0$, $1^\infty$) and those which are more problematic, like $\infty-\infty$ or $\frac\infty\infty$ which cannot be conveniently defined.
First off, nobody ever really means $0^0 = 1$ in a setting where the the exponentiation map $\mathbf{R}_{>0} \times \mathbf{R} \to \mathbf{R}$ is intended.
There is pretty much always something else going on; e.g. meaning to do arithmetic with functions rather than values, and continuously extending the results. That is, in the same settings where you'd write $x/x = 1$ you would also write $x^0 = 1$.
Another common convention is $0 \log 0 = 0$, but that's another example of the same thing: the continuous extension of $x \log x$.
The most widely applicable convention like this I've seen is that of something being "strongly zero". e.g. the Iverson bracket $[P]$ is usually meant to be strongly zero whenever $P$ is false. This means that when $P$ is false $[P]$ multiplied by any expression (even an infinite one or even an undefined expression) still means zero.