Tetration of 0^0^0 .... or h(0)=?

[Ivars]

Note, that the notation 1 - 1 + 1 - 1 + ... - ... can occur as limit of very different powerseries and thus can be assigned an arbitrary value.
The standard case is assuming lim x->1 s(x) or lim x->0 t(x).

I do not think these values are arbitrary. They follow from what series we use to get 1-1+1-1.......... - but no one have shown how they arise.

The problems of the ambiguity of 0^0 when seen as lim x->0 0^x or seen as lim y->0 y^0 are well known, and it is well known, that they don't converge to the same value (which would be necessary for a unique definition).

Gottfried

I believe the problem lies in the idea of limit itself. If ratio between 2 zeros could be any finite value, then of course 0^0 is difficult, except for cases I mentioned, when one of them is infinitely bigger than other- an infinitesimal of another order.

I am not a fan of limits.

But still, what about h(0)? May be that is easier than 0^0 as it involves infinitely many 0 so if their ratio is finite ( they are of same order of infinitesimal) , in infinite exponetiation it either loses its importance or becomes obvious?

And what about the idea that 0^0^0^0 is in fact:

0^1^1^1^1..........* 0^-1^-1^-1^-1............

is that correct?

And than of course 1 can be obtained in many ways, so not all 1 are equal for such exponentiation as well.