1549901369-Elements_of_Real_Analysis__Denlinger_

4.1 Definition of Limit for Functions 179

(4) If Xo is an interior point of V(f), then Definition 4.1.l simplifies to:

lim f(x) = L if Ve> 0, :3 o > 0 3 0 < Ix - xol < o::::} lf(x) - LI < €.

X--+Xo

(5) There is actually a third quantifier here. The universal quantifier on x
is understood to be present, even when left out in the interest of simplicity.

(6) The following statements are interchangeable, and each one will find

use at one time or another:

(a) lim f(x) = L. (b) f has limit Las x --t xo.

x-+xo

(c) f has limit Lat xo. (d) f(x) --t Las x --t xo.

(7) The definitions of lim Xn = L and lim f(x) = L are really quite
n-+oo x-+xo
similar. Much of the development of the theory of limits in this chapter will
closely parallel the theory of limits of sequences developed in Chapter 2. This
chapter is deliberately laid out in a way that helps you see that relationship.

THE c-~ GAME

The process of using Definition 4.1.1 to prove that lim f(x) = L may be
X-+Xo
viewed as an "e-o game." Player #1 chooses an arbitrary€> 0 and challenges
Player #2 (that's you) to find a o > 0 that guarantees that f(x) will be within a
distance€ from L whenever xis within a distance o from xo (without equalling
xo). See Figure 4.1.

y

Givenc. L

Y =f(x)

~

Xo-8

XQ

Find 8

Figure 4.1

x