4.3 One-Sided Limits 205As with Definition 4.3.i, the universal quantifier on x is understood to be
present, even when left out in the interest of simplicity.( 4) The following statements are interchangeable, and each one will find
use at one time or another:(i) lim f(x) = L.
x~xci(ii) f (xt) = L.
(iii) f has limit L as x approaches x 0 from the right.
(iv) f has right-hand limit L at xo.
(v) f has limit L from the right at xo.
(vi) f(x)---+ Las x---+ xt.
lx-21
Example 4.3. 4 Prove that lim --= 1.
x-->2+ X - 2
y(^2) l +t:
Given t:
1- £
-2 -1 (^2) 2+8
f
:io
-2
Figure 4 .8
x
P roof. Let c > 0. Let J = any positive number. Then, 2 < x < 2 + 6 =>
I
Ix -^2 1 - ii= I x -
2
- ii = ji - i j = 0 < c. Thus, lim Ix -
2
1 = 1. 0
X - 2 X - 2 X-->2+ X - 2
LIMIT THEOREMS FOR ONE-SIDED LIMITS
Theorems 4.1.7 through 4.2.23 express the basic facts about the algebra
of limits of functions. Each of these theorems can be revised to express an
analogous fact about one-sided limits of functions. For example,