المحاضرة الخامسة

LIMITS AND CONTINUITY
OVERVIEW Mathematicians of the seventeenth century were keenly interested in the study
of motion for objects on or near the earth and the motion of planets and stars. This study
involved both the speed of the object and its direction of motion at any instant, and they
knew the direction was tangent to the path of motion. The concept of a limit is fundamental
to finding the velocity of a moving object and the tangent to a curve. In this chapter we
develop the limit, first intuitively and then formally. We use limits to describe the way a
function varies. Some functions vary continuously; small changes in x produce only small
changes in ƒ(x). Other functions can have values that jump, vary erratically, or tend to increase
or decrease without bound. The notion of limit gives a precise way to distinguish
between these behaviors.
Rates of Change and Tangents to Curves
Calculus is a tool to help us understand how functional relationships change, such as the
position or speed of a moving object as a function of time, or the changing slope of a
curve being traversed by a point moving along it. In this section we introduce the ideas of
average and instantaneous rates of change, and show that they are closely related to the
slope of a curve at a point P on the curve. We give precise developments of these important
concepts in the next chapter, but for now we use an informal approach so you will see
how they lead naturally to the main idea of the chapter, the limit. You will see that limits
play a major role in calculus and the study of change.
Average and Instantaneous Speed
In the late sixteenth century, Galileo discovered that a solid object dropped from rest (not
moving) near the surface of the earth and allowed to fall freely will fall a distance proportional
to the square of the time it has been falling. This type of motion is called free fall. It
assumes negligible air resistance to slow the object down, and that gravity is the only force
acting on the falling body. If y denotes the distance fallen in feet after t seconds, then
Galileo’s law is
where 16 is the (approximate) constant of proportionality. (If y is measured in meters, the
constant is 4.9.)
A moving body’s average speed during an interval of time is found by dividing
the distance covered by the time elapsed. The unit of measure is length per unit time:
kilometers per hour, feet (or meters) per second, or whatever is appropriate to the problem
at hand.