Is there a path in the complex plane such that $$\lim_{z\to0}z^z\neq1\,?$$
In other words, I would like to approach $(0,0)$ in such a way that $f(z)=z^z$ will result in something different from 1. So far every path I tried gave me 1 as result.
I know that if I pick $y=0$ then: $$\lim_{(x,y)\to(0,0)}y^x=0$$ but I would like a path where $y\neq0$