I would like know real functions $f(x)$ and $g(x)$ for which the following is true:
- $f(x)$ and $g(x)$ are defined when $x \ge 0$
- $\lim_{x \to 0^{+}} f(x)=0$ and $\lim_{x \to 0^{+}} g(x)=0$
- $\lim_{x \to 0^{+}} f(x)^{g(x)}=a \ne 1$
My textbook says that because of functions like this, $0^0$ is not $1$. I think my textbook is wrong and such functions does not exist! Because of (3), $f(x)^{g(x)}$ must be defined (and real) in some neighborhood of $0$, so ia $f(x)= \begin{cases}{} x\sin(1/x) \text{, if }x>0 \\ 0 \text{, if }x=0\end{cases}$ is not valid.
Until today, I thought that reason for not define $0^0$ is that also $0^0=0$ could be "good" definition, along the $0^0=1$. It can be both, so the best solution is not to define.
There's plenty of such functions. I'll consider only $f$ positive, so let $\log f(x)=h(x)$. Your limit then becomes $$ \lim_{x\to0^+}\exp(h(x)g(x)) $$ Note that you want $\lim_{x\to0^+}h(x)=-\infty$. A good choice is then $$ g(x)=x, \qquad h(x)=-a/x \quad(a>0) $$ for which $\lim_{x\to0}\exp(h(x)g(x))=e^{-a}$.
I'll leave to you finding examples where the limit is $\ge1$, $\infty$ or non existent.