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C H A P T E R 3
Limits, Derivatives, Rules, and
the Meaning of the Derivative
Traditional calculus courses begin with a detailed formal discussion of limits and continuity. is
book departs from that tradition, with this chapter introducing limits only in an informal fashion
so as to be able to get going with calculus. A formal discussion of limits and continuity appears
in Chapter 6. e agenda for this chapter is to get you on board with a workable operational
definition of limits; use this to give the formal definition of a derivative; develop the rules for
taking derivatives; and end with a discussion of the physical meaning of the derivative.
3.1 LIMITS
Suppose we are given a function definition like:
f .x/ D
x
2
4
x C 2
en, as long as x ¤ 2, we can simplify as follows:
f .x/ D
x
2
4
x C 2
D
.x 2/.x C 2/
.x C 2/
D
.x 2/
.x C 2/
.x C 2/
D x 2
So, this function is a line—as long as x ¤ 2. What happens when x D 2? Technically, the
function doesn’t exist. is is where the notion of a limit comes in handy. If we come up with
a whole string of x values and look where they are going as we approach 2, they all seem to
be going toward minus 4. e key phrase here is seem to be, and the rigorous, precise definition
of this vague phrase is the meat of Chapter 6.
For now, let’s examine a tabulation of the behavior of f .x/ near x D 2.