Complex analysis vs Real Analysis of $\lim_{x\to0}{x}^{x}$

Before we can start computing $\lim_{x\to0}f(x)$ we have to make sure that $f$ is well defined in a suitable, maybe restricted, neighborhood of $0$. Now powers $a^b$ with arbitrary real (or complex) exponents are bona fide (i.e., without additional explanations) defined only when the base $a$ is a positive real number. In this sense $x^x$ is well defined for $x>0$, and one has $\lim_{x\to0}x^x=\lim_{x\to0}\exp(x\log x)=e^0=1$.

Now it is customary to extend the $\log$ function from the positive real axis ${\mathbb R}_{>0}$ to the slit complex plane ${\mathbb C}':={\mathbb C}\setminus{\mathbb R}_{\leq0}$ by defining the principal value $${\rm Log}\,z:=\log|z|+i\,{\rm Arg}\,z\ ,$$ whereby $-\pi<{\rm Arg}\,z<\pi$ is the principal value of the polar angle of $z\in{\mathbb C}'$. For $a\in{\mathbb C}'$ and arbitrary $b\in{\mathbb C}$ one then defines $$a^b:=\exp(b\,{\rm Log}\,a)\ .$$ This definition extends the definition of "general powers" $(a,b)\mapsto a^b$ from ${\mathbb R}_{>0}\times{\mathbb R}$ to ${\mathbb C}'\times{\mathbb C}$. In this way we can consider $$z^z:=\exp\bigl(z(\log|z|+i\,{\rm Arg}\,z)\bigr)\qquad(z\in{\mathbb C}')\ .\tag{1}$$ While $0\notin{\mathbb C}'$ the origin is certainly a limit point of ${\mathbb C}'$. It is therefore allowed to consider the $\lim_{z\to0}$ in $(1)$. From the well known limit $\lim_{x\to0+}x\log x=0$ it then easily follows that $$\lim_{z\to 0}\bigl(z(\log|z|+i\,{\rm Arg}\,z)\bigr)=0\ ,$$ so that $\lim_{z\to0}z^z=1$. But note that we have excluded the negative real axis completely from the picture. If $\gamma: \ t\mapsto z(t)$ $(0\leq t<\infty)$ is a suitable spiral then using a continuous argument along $\gamma$ the statement $\lim_{t\to\infty}z(t)^{z(t)}=1$ may fail.