The physics relationship of them using averages is:
#u=s/t#
Where #u# is the velocity, #s# is the distance traveled and #t# is the time. The longer the time, the more accurate the average speed can be calculated.
However, although the runner could have a velocity of #5m/s# those could be an average of #3m/s# and #7m/s# or a parameter of infinite velocities during the time period. Therefore, since increasing time makes velocity "more average" decreasing time makes velocity "less average" therefore more precise. The smallest value that time could take would be 0, but that would nulify the denominator. Therefore, one uses the limit as #t# tends to, but never approaches, 0.
#u=lim_(t->0)(s/t)#
