sequences and series - Yet another nested radical

Consider $$F(x) = \sqrt{x -\sqrt{2x - \sqrt{3x - \cdots}}}$$

I believe I can prove (with some handwaving) that

• $F$ does converge everywhere in $\mathbb{C}$


• $\Im F = 0$ for sufficiently large real $x$ (actually larger than $x0 \approx 0.5243601\dots$ Does this number ring a bell?)


• Coincidentally $F(x0) = 0$

Weird things happen in the limit to $0$. Obviously, $F(0) = 0$. However, it seems that $$\lim_{x \to +0}F(x) = \overline{\zeta}$$ $$\lim_{x \to -0}F(x) = \zeta$$
where $\zeta = \frac{1 + i\sqrt{3}}{2}$ is a usual cubic root of $-1$. Moreover, $F$ seems to reach one of those as $x$ approaches $0$ at a rational angle. I understand that this may well be a computational artifact (still making no sense to me), but proving or refuting these limits is definitely out of my league.

Any help?