1549901369-Elements_of_Real_Analysis__Denlinger_

4.2 Algebra of Limits of Functions 191

Lemma 4.2.10 (Fundamental Limit) For every x 0 E IR., lim x = x 0.

Proof. Exercise 3. •

Theorem 4.2.11 (Algebra of Limits of Functions)

Suppose lim f(x) = L, lim g(x) = M, and c ER Then

X-+Xo X-+Xo

(a) lim cf(x) = cL;

X-+Xo

(b) lim (f(x) + g(x)) = L + M;

X-+Xo

(c) lim (f(x) - g(x)) = L - M;

X-+Xo

(d) lim (f(x) · g(x)) = LM;

X-+Xo

(e) lim ·(-

1

-) = 2_ (if M ::/= O);

x->xo g(x) M

( f) lim ( f ( x) ) = .!:_ (if M ::/= 0);

x->xo g(x) M

X-+Xo

(g) lim J7(x) =VI (if f(x) 2: 0 for all x in some N§ (xo)).

x-+xo

[In (b), (c), (d), and(!), we assume that xo is a cluster point of V(f) n
V(g).]

Proof. Suppose lim f(x) = L, lim g(x) = M, and c ER Then
X-+Xo X--+Xo
(a) Case 1 (c ::/= 0): Let c > 0. Since lim f(x) = L, 3 o > 0 3 't:/x E V(f),
X-t-Xo
€
0 < Ix -xol < o ::::?-lf(x) -LI < ~-Then, 't:/x E V(f),

0 <Ix -xol < O :::? lcf(x) -cLI = lcllf(x) -LI

Case 2 (c = 0): Exercise 4.

(b) Let c > 0.

€

< lcl·-

lcl

< €.

€
Since lim f(x) = L, 3 01 > 0 3 't:/x E V(f), 0 < lx-xol < 01:::? lf(x)-LI < -
2
.
x-+xo
€
Since lim g(x) = M, 302 > 0 3 't:/x E V(g), 0 < Ix - xol < 82:::? lg(x)-MI < -
2
.
x-+xo