Angular velocity as $r \to 0$ I know that angular velocity is defined as the rate of change of angular displacement and is related to tangential velocity via $$\omega = \frac{r\times V}{|r^2|}$$
My question is what will be the value of angular velocity at the centre of rotation i.e. $$\lim_{r \to 0} \omega = ?$$
 A: The center of the rotation is a singular point of the motion. There, the tangential velocity field, $\vec{v}(r)$, is not well defined. It's the same singularity that is present in the unit vectors of the polar coordinate system that comes from the fact that at $r=0$, $\theta$ can have any value and still refer to the same point. 
This singularity is typically plugged by taking the limit in the question, though, in the same way that, strictly speaking, $\frac{x}{x}$ is not defined for $x=0$ (for a more practical application of this strategy, see the sinc function). The limit does exist, because in rotations of solids for every $r > 0$, $\vec{\omega}$ is a constant.
A: The angular frequency remains finite. Consider a rigid body rotating such that each point rotates an angle $d\theta$ in the time $dt$. The angular velocity (and what is usually termed angular velocity) is given by:
$$\omega=\frac{d\theta}{dt}$$
which is constant for all points on the rigid body. In the formula you have given as $r\rightarrow 0$, $V\rightarrow 0$ like $r$ thus you get an expression of:
$$\omega\propto \lim_{r\rightarrow 0}\frac{r^2}{r^2}$$
which is finite and will correspond to the value of $d\theta/dt$.
