NOTE: I moved this question here from MathOverflow in accordance to suggestions I received there.
It's been many years since my university days, but there is one thing that my mathematics professor once said, which has stuck with me all this time.
I no longer remember the exact context, but I do remember him saying something along these lines (heavily paraphrased from memory):
Imagine you are driving your car and you see a red traffic light in front of you. You're still 50 or so meters out so you start braking gently, trying to bring the car to a complete stop just at the traffic lights.
Now came the twist:
As soon as you are happy with the rate of deceleration, you close your eyes and keep them closed until you feel the stop.
Bear in mind that he was in no way encouraging us to actually try this! It was merely a thought experiment of sorts.
He then went to explain the point of the exercise:
When the car stops, you will feel an abrupt point. The strength may vary with the rate of deceleration, but you are bound to feel it in most situations because you've eliminated most of the distractions by closing your eyes. And when it happens, it will quite probably scare you into opening your eyes!
The abrupt stopping point is as close as you will get to experience a division by zero in real world. It occurs when function
f(v) = dx / dtapproaches its limit and then finally jumps to0/0.
Since then, I've seen many paradoxical equations that all basically come from a misguided attempt at division by zero and I understand why only nonsense can follow any such step.
But I'm still wondering... Is there any truth to what my prof said? Is velocity changing from something non-zero to zero actually an example of division by zero in the real world, maybe related to discrete vs. continuous model of real world?
This is not just wrong, it’s actually, if we want to attach any sense at all to the non-rigorous concept of “dividing zero by zero”, the wrong way around.
First, motion is relative; there's no such thing as coming to rest. You come to rest relative to the street. If something as deep and metaphysical as you describe would happen when you stop on the street, that would establish the street as a distinguished frame of reference, contradicting all of our current physics.
Second, if it makes any sense at all to speak of “dividing zero by zero” then that’s what we always do to calculate your velocity. Velocity is defined as the limit of the ratio of two quantities that go to zero: the distance travelled in a short time interval over the time interval. The one case where you don’t need such a construction to define a velocity is when an object is at rest in some inertial frame (in this case, neglecting the Earth’s rotation, the street), because then the ratio doesn’t depend on the time interval and we don’t have to take a limit to calculate it.
Thus, what you feel abruptly may be your relief at having a well-defined velocity without having to go through an esoteric limiting process to establish it.
On a more serious note: There are several things wrong with this. There’s no such thing as 0/0 to jump to. The “approaching the limit” doesn't occur in time, it occurs in an abstract process unrelated to time, so it makes no sense that the velocity should “jump” to the limit at some point in time. There are also trivial errors, e.g. that it should say $v$, not $f(v)$. This is all one big chunk of nonsense, and your professor apparently had no idea what he was talking about (or he was pulling your leg or you’re misremembering him). I tried to convey how nonsensical it is by my somewhat humourous answer.