What right do we have to take the number approached by the average speeds to be the speed at t = 4? The answer is that mathematicians have adopted a definition which makes good physical sense. They argue that the smaller the interval of time bordering t = 4 over which the average speeds are computed, the closer must the behavior of the falling body be to that at t = 4. Hence the number approached by average speeds over the smaller and smaller intervals of time bordering t = 4 should be the speed at t = 4. Since mathematics seeks to represent physical phenomena, it quite naturally adopts
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