Is $x^{-0}$ defined? In the form of mathematics that most of humanity is taught, the following operation is undefined:
$\Large{\frac{x}{0}}$
But, how about the following operation?
$\Large{x^{-0}}$
Is the following statement true?
$\Large{x^{-0}=\frac{1}{x^0}=\frac{1}{1}=1}$

There are two principles: "anything to the power $0$ is $1$" and "$0$ raised to any positive power is $0$". It makes sense to allow the first principle to define $0^0:=1$, because there is no reason for the second principle to extend to $0^0$, as it clearly does not extend to any negative power, and there are abundant examples, such as $\lim_{x\to0+}x^x=1$, which would violate it. That said, one can easily find examples where such a limit may not be $1$: Consider $\lim_{x\to0+}y^x$, which is $0$ in the case that $y=\exp(-1/x^2)$ although both $x$ and $y$ go to $0$. It is probably because of cases like this that some mathematicians are uncomfortable with the convention that $0^0=1$, and prefer to consider it undefined. 
