1549901369-Elements_of_Real_Analysis__Denlinger_

198 Chapter 4 • Limits of Functions

Theorem 4.2.22 (Limits Preserve Inequalities)

(a) If lim f(x) = L and f(x) ::::: K for all x in some deleted nbd. of xo, then

X-+Xo

L :SK.

(b) If lim f(x) =Land f(x);::: K for all x in some deleted nbd. of xo, then

X--+XQ

L;:::K.

(c) If lim f(x) and lim g(x) exist, andf(x) :::;g(x)forallxinsomedeleted

X-+Xo X-+Xo

nbd. of Xo, then lim f(x) :S lim g(x).

X-+Xo X-+Xo

Proof. (a) Suppose lim f(x) =Land 381 > 0 3 \:/x E Nfi (xo), f(x) :S
x--+xo 1
K. We want to prove that L :S K. For contradiction, suppose L > K. Let
c = L - K. Then c > 0. Since X--+Xo lim f(x) = L, 382 > 0 3 \:/x E Nfi 2 (xo),

lf(x) - LI< c. Let 8 = min{81,82}. Then,

x E Nfi(xo) ::::} x E Nfi 2 (xo) and x E Nfi 1 (xo)

::::} lf(x) - LI < c and f(x) :SK

Therefore, L ::::: K.

(b) Exercise 1 7.

::::} - c < f(x) - L < c and f(x) :SK

::::} L - c < f ( x) < L + c and f ( x) :S K

::::} L - (L - K) < f(x) and f(x) :SK

::::} K < f(x) and f(x) :SK. Contradiction!

(c) Suppose lim f(x) =Land lim g(x) = M, and f(x) :::; g(x) for all x
X--+Xo X-t-Xo
in some deleted nbd. of xa. Define the function h(x) = g(x) - f(x). Then, by
the algebra of limits theorem,

lim h(x) = lim g(x) - lim f(x).

X--+Xo X-+Xo X-+Xo

Now, \:/x-=/= xo, h(x);::: 0, so by Part (b) above, lim h(x);::: 0. That is,

x--+xo

lim g(x) - lim f(x);::: O; i.e.,

X-+Xo X--+Xo

lim g(x);::: lim f(x). •

x--+xo x--+xo