1549901369-Elements_of_Real_Analysis__Denlinger_

188 Chapter 4 • Limits of Functions

we look for the implications of 0 < Ix - xol < 8 instead of n 2: no. As you go

through this section, you are encouraged to observe how this section parallels

Section 2.2. That will help you learn and remember these ideas.

Theorem 4.2.1 (Absolute Value and Limits)

(a) X---+Xo lim l(x) = 0 {:} X-+Xo lim ll(x)I = O;

(b) lim l(x) = L {:} lim ll(x) -LI= O;

x-+xo x-+xo

(c) lim l(x) = L =? lim ll(x)I = ILi, but the converse is not true.

X---+Xo X---+Xo

Proof. Exercise l. •

SOME TERMINOLOGY USEFUL FOR LIMITS

Some additional language and terminology of sets and functions will be

useful. We review the basic terminology^6 here, and relate it to the theory of

limits.

Definition 4.2.2 Suppose 1 : D(j) _, JR is a function, A ~ D(j) and B ~ JR.

Then

l(A) = {l(x) : x EA};

l-^1 (B) = {x: l(x) EB}.

The set f(A) is called the image of A under f and the set l-^1 (B) is called

the inverse image of B under 1.

Definition 4.2.3 (Deleted Neighborhoods) Let x 0 E JR. Ve> 0, we define

the deleted €-neighborhood of x 0 to be the set (see Figure 4.3)

N~(xo) = Ne(xo) - {xo}

= {x:O<lx-xol<c}

= (xo -c, xo) U (xo, xo + c).

6. For a review of the language and terminology of sets and functions, see Appendices B .2
   and B .3.