Why Read This Book? - Index of

7.2 Limits and Their Basic Properties 213
at that in Section 7.5. There are plenty of examples in which a hole in the domain
occurs naturally, and we will look at one example later in this section.
7.2.2 Basic Theorems of Limits
There is a real similarity between the definition of convergence of a sequence as
n →∞and convergence of a function as x → a. In the convergence of sequences,
a given ɛ>0 must be associated with a threshold term beyond which all terms of
the sequence fall in the ɛ-neighborhood of the limit. In the convergence of a function
f , a given ɛ>0 must be associated with a δ-radius of a deleted neighborhood
of a, within which all values of the function, except perhaps f(a), must fall in the
ɛ-neighborhood of the limit. The good news is that the similarity of these definitions
makes for plenty of similar theorems, with similar proofs to match. So
here is a barrage of theorems involving convergence of a function. As we present
them, we will often link them back to sequence theorems and properties. We prove
some of them here to illustrate the similarities, but you will prove most of them
as exercises by mimicking your work from Section 6.2.
Theorem 7.2.4 For any real numbers a and c, the constant function f(x) = c
satisfies limx→a f(x) = c.
Proof. Let ɛ>0 be given, and let δ = 1. Then if 0 < |x − a|