Schwarzschild metric - minimum rate of decreasing $r$

So in the Schwarzschild metric, I want to find the minimum rate at which the coordinate $r$ decreases for a massive particle wrt to the proper time (Ex 5.3 Carroll). The particle doesn’t need to be in a geodesic and I consider the particle inside the horizon of the BH. So I used the massive particle condition:

$$U^2=-1$$
and I got:

$$left(frac{d r}{d tau}right)^{2}=left(frac{2 G M}{r}-1right)left[1+left(frac{2 G M}{r}-1right)left(frac{d t}{d tau}right)^{2}+r^{2}left(frac{d Omega}{d tau}right)^{2}right]$$
So now I can argue that motion along $theta$ or $phi$ is gonna increase the rate. So to find the minimum one I consider a radial motion:

$$left(frac{d r}{d tau}right)^{2}=left(frac{2 G M}{r}-1right)left[1+left(frac{2 G M}{r}-1right)left(frac{d t}{d tau}right)^{2}right]$$
At this point I would have the minimum rate for

$$left(frac{d t}{d tau}right)^{2} rightarrow 0 $$
But what is the meaning of this limit? Can I actually take it?