0^0 = 1?

[AlexFekken]

No actually, I am not moving goalposts: I am applying the standard mathematical definition of a limit for a function of two variables. And rather than "being defined from all directions" not being necessary, this definition is even stronger than that (based on the usual epsilon-delta or topological criteria).
Limits for functions of a single (real) variable are deceptive because there is only "from above" and "from below". In 2 and more dimensions "all directions" do need to be taken into account to get a decent definition of "limit".

With regards to you log(x) example, that only reinforces my statement that the (topology and) domain choice for the variables (arguments) matters. If you only define log(x) for x>0, and you extend the real numbers with +Inf and -Inf, then you are right. If you don't extend the real numbers then there is no limit ("out of range" if you like). But if you extend the definition of log(x) to x<0 then there is no limit either. Where you say "undefined" (for x<0), I could say "out of range" because you failed to extend the real numbers to the complex numbers.

You could also extend the real numbers with (unsigned) Inf only and then 1/x would also have a limit. This may seem silly for real numbers but it is in fact standard practice for complex numbers and 1/0 is in fact treated as equal to Inf in the extended complex plane, i.e. on the Riemann sphere.

But the case for 0^0 is much simpler because you can stick to positive numbers to see why it should not be defined; unless you limit yourself to integer exponents.