Defining 0^0 = 1 is just as important in analysis as it is in combinatorics, even though analysis texts aren’t always clear that they’re making use of this fact.
• If p(x) = ∑[n=0..∞] a_n x^n is a power series, then p(0) = a_0 is its constant term, but it makes no sense to write this unless 0^0 = 1.
• The power rule d/dx [x^n] = n x^(n−1) holds for n = 1 and x = 0, but this requires 0^0 = 1.
The reason that some people believe it’s important to undefine 0^0 in analysis is that the limiting expression
lim[x → a] f(x)^g(x)
does not necessarily exist when lim[x → a] f(x) = lim[x → a] g(x) = 0. But all this means is that we have to draw a distinction between the _value_ 0^0, which equals 1, and the indeterminate _limiting form_ 0^0, which is an abbreviation for the above type of limit.
It is not uncommon for values to evaluate differently from the corresponding limiting forms. For example, the value floor(0) equals 0, but the limiting form floor(0) is indeterminate. It may seem surprising that such a discrepancy arises for exponentiation, but all it means is that exponentiation is discontinuous at (0, 0), as it must be.
(Note however that the above limit _does_ exist with mild conditions on f and g: if f, g are complex analytic functions with f not identically zero, then the limit equals 1.)
Defining 0^0=1 has issues. Example: a(n)=1/exp(n), b(n)=1/n. Then both a and b converge to 0, but a^b is always exp(-1) (and hence does not converge to 1).
What you have shown is that the _limiting form_ 0^0 is not always equal to 1. The _value_ 0^0 is still equal to 1.
Similarly, floor(−1/n) converges to −1, which tells us that the _limiting form_ floor(0) is not always equal to 0; but the _value_ floor(0) is still equal to 0. Nobody uses this to argue that the value of floor(0) should be undefined or context-dependent.
But you are right, what I meant is there is no way to define 0^0 maintaining continuity of the power function. Why is this important? Because power is a continuous function otherwise.
Similarly, there is no way to define floor(n) for integers n maintaining continuity of the floor function, even though floor is a continuous function otherwise. We still define floor(n) = n because the meaning of the floor function is more important than its continuity. And so it is with exponentiation at (0, 0).
• If p(x) = ∑[n=0..∞] a_n x^n is a power series, then p(0) = a_0 is its constant term, but it makes no sense to write this unless 0^0 = 1.
• The power rule d/dx [x^n] = n x^(n−1) holds for n = 1 and x = 0, but this requires 0^0 = 1.
The reason that some people believe it’s important to undefine 0^0 in analysis is that the limiting expression
lim[x → a] f(x)^g(x)
does not necessarily exist when lim[x → a] f(x) = lim[x → a] g(x) = 0. But all this means is that we have to draw a distinction between the _value_ 0^0, which equals 1, and the indeterminate _limiting form_ 0^0, which is an abbreviation for the above type of limit.
It is not uncommon for values to evaluate differently from the corresponding limiting forms. For example, the value floor(0) equals 0, but the limiting form floor(0) is indeterminate. It may seem surprising that such a discrepancy arises for exponentiation, but all it means is that exponentiation is discontinuous at (0, 0), as it must be.
(Note however that the above limit _does_ exist with mild conditions on f and g: if f, g are complex analytic functions with f not identically zero, then the limit equals 1.)
See also Donald Knuth’s _Two notes on notation_: http://arxiv.org/abs/math/9205211.