When you enter a darkened room, your eyes adjust to the reduced level of light by increasing the size of your pupils. Enlarging the pupils allows more light to enter the eyes, which makes objects around you easier to see. By contrast, when you enter a brightly lit room, your pupils contract, reducing the amount of light entering the eyes. This is necessary since too much light will overload your visual system.
This visual adjustment mechanism is present in many animals. Researchers study this mechanism by performing experiments and trying to find a mathematical description of the results. In this case, you might want to represent the size of the pupils as a function of the amount of light present. Two basic characteristics of such a mathematical model would be
Finding a function with these two properties can be a challenge. (Try it!) One possible graph of such a function is shown in Figure 1.1. (See example 5.11 for more.) In this chapter, we develop the concept of limit, which can be used to describe functions with specific properties such as those listed above. The limit is the fundamental notion of calculus. This underlying concept is the thread that binds together virtually all of the calculus you are about to study.Aninvestment in carefully studying limits now will have very significant payoffs throughout the remainder of your calculus experience and beyond.
- As the amount of light (x) increases, the pupil size (y) decreases down to a minimum value p; and
- As the amount of light (x) decreases, the pupil size (y) increases up to a maximum value P.