exponential function - Why is $0^0$ undefined?

I'm wondering why $0^0$ is considered undefined. Why isn't 1 considered a valid solution?

Considering $0^0 = 1$ seems reasonable to me for two reasons:

1. $\lim_{x \rightarrow 0} x^x = 1$




2. $a^x$ would be a continuous function

Could you please explain why 1 can't be a solution and maybe provide some examples that show why having $0^0$ undefined is useful?

Answer

0Because as a function $f(x,y): R^2 \rightarrow R = x^y$ we have two different values moving toward $f(0,0) = 0^0$. In other words, $f(0^+,0) = 1$ and $f(0,0^+) = 0$.

But beware that there are some places in mathematics which by convention accept one of these values. For example in some parts of combinatorics we have $0^0 = 1$ to ease the definition of some functions.