How do you get the magnitude of the derivative of the unit vector in the limiting process?

For the $\Delta t \rightarrow 0$ limiting case for $\dot{\hat{r}}$, the magnitude is $\frac{\Delta \hat{r} }{\Delta t}$; $\Delta \hat{r}$ is $\vert\hat{r}\vert \Delta \phi$ (arc length), but since $\hat{r}$ is a unit vector, $\vert\hat{r}\vert =1$, so $\frac{\Delta \hat{r} }{\Delta t} \sim \frac{\Delta \phi}{\Delta t}$, which in the limit is $\dot{\phi}$.

For the $\Delta t \rightarrow 0$ limiting case for $\dot{\hat{\phi}}$, see above question.