How do I rigorously compute $$\lim_{x\rightarrow0} a^x$$ for $a \in \mathbb{R}$?
I can intuitively and graphically get the answer of $\delta_{a\neq0}$ (Kroenecker delta), and I think also by using the $(\delta, \epsilon)$ definition, but I'm not sure if I'm doing it hand-wavely or rigorously.
UPDATE: the reason I am asking is because I'm thinking about teaching in statistics about median, mean, and mode, and how these quantities minimize a sum of distances $d_n(x−m)=|x−m|^n$, where for $n=2$ you get the mean, for $n=1$ you get the median, and, under this definition above, for $lim_{n\rightarrow0}$ you get the mode; but I wanted to make sure my math is rigorous in case I get fielded questions from students.
Another consideration is to use $a^x = e^{x \, \ln(a)}$ and $$ e^{t} = 1 + t + \frac{t^2}{2!} + \frac{t^3}{3!} + \cdots $$ to obtain the limit \begin{align} \lim_{x \to 0} \, a^{x} &= \lim_{x \to 0} \, e^{x \, \ln(a)} \\ &= \lim_{x \to 0} \left(1 + x \, \ln(a) + \frac{x^2 \, \ln^{2}(a)}{2!} + \frac{x^3 \, \ln^{3}(a)}{3!} + \mathcal{O}(x^4) \right) \\ &= 1 \end{align} for $a>0$. In the case $a$ is negative, say $a = -|b|$, then $\ln(a) = \ln(e^{\pi i} \, |b|) = \ln(|b|) + i \, \pi$ and leads to the same limiting result. The remaining case of $a = 0$ can also be considered and the $(\delta, \epsilon)$ definitions could also be applied.