How to find instantaneous and average velocity?

Average velocity is equal to the change in distance / change in time :

v(avg) = s / t

that's the average distance, but at some point the object could have moved faster / slower and we want to see the instantaneous velocity. The principal is the same. You take a a little piece of the road for a little piece if time and get the avg velocity of the tiny part of the road. But this way it will be not the exactly speed at that moment, it will be just the avg speed in the little fragment. So we must add limits. What will be the velocity for so tiny part of time that it approaches zero?
We get :

lim s/t = ds/dt
t->0