1549901369-Elements_of_Real_Analysis__Denlinger_

190 Chapter 4 • Limits of Functions

Definition 4.2.8 A function f is said to be bounded away from^7 0 on a

set A~ V(f) if :JC> 0 3 \:/x EA, lf(x)l 2: C.

Theorem 4.2.9 Suppose lim f(x) = L =f. 0. Then there is a deleted neigh-

x-+xo

borhood N8(xo) such that f is bounded away from 0 on N8(xo) n 'D(f). In

fact,^8

(a) If 0 < C < L , then 3 8 > 0 3 \:/x E 'D(f), 0 < Ix - xol < 8::::} f(x) > C.

(b) If L < C < 0, then 3 8 > 0 3 \:/x E 'D(f), 0 < Ix - xol < 8::::} f(x) < C.

(c) IfO < C < ILi, then 38 > 0 3 \:/x E 'D(f), 0 < lx-xol < 8::::} lf(x)I > C.

Proof. (a) Suppose li m f(x) = L =f. 0, and 0 < C < L (see Figure 4.4).

X-+Xo

Then L - C > 0. By Definition 4.1.1 with t: = L - C, 3 8 > 0 3 \:/x E 'D(f),

0 < Ix - xol < 5 ::::} lf(x) -LI< L - C

::::} C - L < f(x) - L < L - C

::::} C < f(x) < 2L - C

::::} f(x) > C.

y

Figure 4.4

Thus, x E N8(xo) n 'D(f) ::::} f(x) > C.

The proofs of (b) and ( c) are similar. •

7. For an extension of this idea , see Exercise 20.

Y =f(x)

x

8. You are encouraged to express these statements less formally, in words, to you rself. For
   example, read (a) as, "If C is less than L, then f(x) is greater tha n C on some deleted
   neighborhood of xa."