To appreciate what the limit process achieves we might consider an analogy. Suppose that a marksman seeks to hit a particular spot on a target. Even if he is a good shot, he is not likely to hit the given spot squarely, but will hit all around it and indeed come close. A bystander observing the location of the hits will readily determine the exact spot at which the marksman is aiming, by noting the concentration of the hits. This process of inferring the precise location at which the marksman is aiming is analogous to determining the instantaneous speed from a knowledge of the average speeds.
...more
