Chapter 4
Limits of Functions
Limits are basic to analysis. But we have done much of the
hard work in Chapter 2. In this chapter we carry over the
tools developed there to the limits of functions. We shall en-
counter very little difficulty and only a few new ideas. The
basic E:-0 techniques, discussed in Section 4.1, are extremely
important. The sequential criterion is shown to be very use-
ful. Sections 4.2 and 4.4, on the algebra of limits, parallel
Sections 2.2- 2.3 closely. One-sided limits are covered in Sec-
tion 4.3.4.1 Definition of Limit for Functions
As you will recall, the idea of limits of functions underlies the entire subject
of calculus. Without an understanding of limits, the concepts of derivative
and integral cannot be made rigorous. Thus, the subject of this chapter is of
fundamental significance. What may be a new insight for you is that the theory
of sequences plays an important role in the theory of limits of functions.
Our first task is to define the statement lim f(x) = L. To help our defini-
x-.x0
tion make sense, we reflect a little on the intuitive notion of limit. First, a word
about notation. To indicate that f is a real-valued function^1 with domain V(f),
we write f : V(f) ----> R Remember that in taking the limit "as x approaches
xo,'' we do not really care about the value of f (xo), nor even whether f (xo)
exists. We only care about f(x) for values of x "close to" (but different from)
x 0. Thus, we do not require that x 0 be in the domain off. But we do require
that x 0 be a cluster point of the domain- otherwise values of x in the domain
- For a review of the fundamental ideas and notation of functions, see Appendix B.
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