Why limits?
Math 120 Calculus I
D Joyce, Fall 2013
Why limits? We’ll spend the next few weeks studying “limits.” Naturally, the question is
“why limits?” Why not just go on to derivatives?
The answer involves the character of the course. This is not just a course about how to use calculus, but a mathematics course about what calculus is.
How do derivatives depend on limits? Just how are derivatives supposed to depend on limits? A derivative is supposed to be the rate of change of a function at an instant, what we’ll call “instantaneous rate of change.” On the face of it, that makes no sense at all, since an instant is a point in time, and nothing changes at a point in time. You need a time interval for anything to change.
Start with a function y = f (x). To help us understand, let’s take x to be time, measured in some convenient time unit, and let’s take y = f (x) to be the distance travelled at time x, measured in some convenient distance unit. Then the derivative is what we know as velocity.
The velocity doesn’t have to be constant, but may change over time. It might slowly at first with a small velocity, and later quickly with a large velocity, or vice versa. We’re trying to determine how to find the velocity f (x) when we know the distance f (x). Finding the derivative f (x) when you know f (x) is called differentiation.
Average rates of change and slopes of secant lines. We can fairly easily compute the average rate of change, that is, the average velocity, over an interval.
(b,f (b))

q

(a,f (a))

q

y = f (x)

a

b

Suppose we take the time interval [a, b] which starts at time x = a and ends at time x = b.
We can compute the distance travelled over that interval as the difference f (b) − f (a). But the length of the time interval is b − a. That says the object travels a distance of f (b) − f (a) units over a time interval of