178 Chapter 4 • Limits of Functions
off could not get "close to" x 0. Now, saying that f(x) gets close to Lis saying
that lf(x) -LI gets small. Similarly, saying that x gets close to (but not equal
to) x 0 is saying that Ix - x 01 gets small without equaling 0. Now, we are ready
for the definition.
Definition 4.1.1 If f: 'D(f) ~JR, and x 0 is a cluster point of 'D(f), then
lim f(x) = L if
X-+Xo
Ve> 0, :35 > 0 3 "Ix E 'D(f), 0 <Ix -xol < 6 =? lf(x) - LI< e.That is , lim Xn equal to L means:
n->oo
lf(x) -LI is arbitrarily small whenever x E 'D(f) is sufficiently close to xo
(but not equal to Xo),
or
Ve > 0, there is some 6 > 0 such that lf(x) - LI < e whenever x E 'D(f) and
0 <Ix - xol < 6. D
Verbal Paraphrase of Definition 4.1.1.^2
lim f(x) = L {o} f(x) can be made arbitrarily close to L by making
X--+Xo
x E 'D(f) sufficiently close to x 0 (but not equal to x 0 ).Notes on Definition 4.1.1:
(1) If :JL E JR 3 lim f(x) = L , we say that lim f(x) exists; otherwise,
X--+Xo X--+Xo
we say that lim f(x) does not exist.
X-+Xo
(2) We shall never say that lim f(x) exists unless x 0 is a cluster point of
X--+Xo
'D(f). For example, lim v,x"""""^3 --x...,,.^2 does not exist. [The domain of the function
x->O
j(x) = Jx^3 - x^2 is {O} U [1, + oo), and 0 is not a cluster point of this set.]
(3) Even if Xo E 'D(j), the value off (xo) is irrelevant to the consideration
of whether lim f(x) = L. The condition "O < Ix - x 0 1" in Definition 4.1.l
X--+XQ
guarantees that we are not letting x = xo.
- Although Definition 4.1.1 is officially correct, and should b e memorized, it is equally im-
portant (for the sake of understanding) to b e able to paraphrase it in words.