How is $0^x$ defined (seriously)?

Question

Normally, $a^b$ is defined as $e^{b \ln(a)}$ and $e^x$ is defined by its Maclaurin Series, and $\ln(x)$ is defined as the real inverse of $e^x$. If we try this with $0^x$, we get $e^{x \ln(0)}$, which is obviously incorrect. Of course we can just define it separately, and say $0^x = 0$ for $x \gt 0$ (or $\geq$ 0 if you're feeling spicy), but this feels inelegant.

Answer 1

Use the Maclurin series you mentioned to find that $$\lim_{\delta \to 0^+}\delta^x=0$$ and hence $0^x=0$ (of course for positive $x$ only).

Does that help?