0^0 = 1?

[Thomas Okken]

I'm sure you can observe all sorts of interesting behaviors in lim (x->0) f(x)^g(x), but that's not helping with the question that started this thread; that's just changing the subject.

Regarding this new requirement that in order for a limit to be defined, it must be defined from all directions: that's called "moving the goalposts." :-) In order for a limit to be defined, it must be unique, but being defined from all directions is not necessary.

Consider: lim (x->0) 1/x is undefined, because the limit from above is +Inf, while the limit from below is -Inf. Not equal, therefore undefined.

Now consider lim (x->0) log(x). From above, -Inf; from below, undefined. Would you say that lim (x->0) log(x) is not -Inf?