When can $0^0=0$?

Question

Here I can see that $$\lim_{x\to 0} x^x=1$$

if it is the case that $0^0$ is undefined except through limits what would an example be of an equation where the limit is $0$ or another value?

Answer 1

Examples ... all limits are as $x \to +\infty$ \begin{align} u &= e^{-x} \qquad v= \frac{1}{x^2}\; \qquad \lim u =0 \qquad \lim v =0 \qquad \lim u^v = 1 \\ u &= e^{-x} \qquad v= \frac{1}{x}\; \qquad \lim u =0 \qquad \lim v =0 \qquad \lim u^v = e^{-1} \\ u &= e^{-x} \qquad v= \frac{1}{\sqrt{x}} \qquad \lim u =0 \qquad \lim v =0 \qquad \lim u^v = 0 \\ u &= e^{-x} \qquad v= \frac{a}{x}\; \qquad \lim u =0 \qquad \lim v =0 \qquad \lim u^v = e^{-a} \end{align}

Answer 2

Let $f(x)=0$, $g(x)=\vert x\vert$. Then $\lim_{x\rightarrow 0}f(x)^{g(x)}=0$ (since for all $a\not=0$ we have $f(a)^{g(a)}=0$).

Answer 3

See two examples page 9 in the paper : "Zero to the zero-th power" https://fr.scribd.com/doc/14709220/Zero-puissance-zero-Zero-to-the-Zero-th-Power . In one example, the limit is $e^2$. In the other example, the limit is $e^{-1}$. One can find as many examples as wanted with different values of limit.