It seems like this limit converges to $\frac{\sqrt3}2$, which matches up with experimental values. While I don't have a formal proof for that, the idea is that $\sin(x) = x - \frac{x^3}6 + O(x^5)$, so we start with $\frac1{\sqrt n}$ and repeatedly subtract $\frac{x^3}6$. We can approximate this discrete system with the differential equation $f'(x) = -\frac{f(x)^3}6$, which has a solution $f(x)=\frac{\sqrt3}{\sqrt{c + x}}$, and we want $f(0) = \frac1{\sqrt n}$, so $c = 3n$, and then $f(n) = \frac{\sqrt3}{2\sqrt n}$.
I believe this can be formalized using ideas from dynamical systems, but I don't know how to do that.