206 Chapter 4 11 Limits of Functions
Theorem 4.3.5 (Sequential Criterion for One-Sided Limits of Func-
tions)
(a) lim f(x) = L {::}\;/sequences {xn} in D(f) n (-oo,xo) 3 Xn --+ Xo,
X-+XQ
f(xn)--+ L.(b) lim f(x) = L {::} \;/ sequences {xn} in D(f) n (xo, oo) 3 Xn --+ Xo,
x-+xci
f(xn)--+ L.(Compare with Theorem 4.1.9.)Theorem 4.3.6 (a) If lim f(x) =LE JR, then f is bounded on some interval
x-+xQ
of the form (xo - 6, xo) where 6 > 0.
(b) If lim f(x) = L E JR, then f is bounded on some interval of the form
x-.xci
(xo, xo + 6) where 6 > 0.(Compare with Theorem 4.2.7.)Theorem 4.3.7 (Limits from the Left Preserve Inequalities)
(a) If lim f(x) = L and ::io > 0 3 f(x) ::::; K for all x E (xo - 6, xo) n D(f),
x-+x(J
then L::::; K.(b) If lim_ f(x) =Land ::36 > 0 3 f(x) 2: K for all x E (xo -6,xo) nD(f),
X-+Xo
then L 2: K.(c) If lim f(x) and lim_ g(x) exist, and ::3 6 > 0 3 f(x) ::::; g(x) for all
x-+xQ x-+x 0
x E (xo - 6,xo) in D(f) n D(g), then lim f(x)::::; lim g(x).
x-+xQ x-+x()(Compare with Theorem 4.2.22.)LIMITS VS. ONE-SIDED LIMITSThe following theorem expresses an important relationship between limits
and one-sided limits.
Theorem 4.3.8 If xo is a cluster point of D(f) n(-oo, xo), and a cluster
point of D(f) n(xo, oo), then lim f(x) = L {::} both lim f(x) = L and
X-+Xo X-+XQ
lim f(x) = L.
x-+xci