Given that expansion had started , there would be no need for an infinity result since speed is distance divided by time, and, for speed, it wouldn't be a set time but the difference in the two points of time for the distance traveled, so in this case, it would be t1 - t0, a value not zero.
But, your infinity point is still a good one because what if t1 is a value super incredibly small, then the speed would necessarily be incredibly fast, faster than light no doubt. So, to the point I keep trying to make, the laws of physics completely breaks down before reaching t=0, and what you note is one reason.
The theory called Big Bang, for infinite reasons, does not begin at t=0. You need metaphysics to make clever suppositions to address energy, density, etc. at t=0, and these suppositions are outside the magisteria of formal science, which is objective based.