1549901369-Elements_of_Real_Analysis__Denlinger_

194 Chapter 4 • Limits of Functions

Proof. (Alternate) Suppose {xn} is a sequence in [D(f)nD(g)]-{xo} 3
Xn ---; x 0. Since lim f(x) = L, the sequential criterion for limits of functions
X--+Xo
(Theorem 4.1.9) guarantees that f(xn) ---; L. Since lim g(x) = M, the se-
x--+xo
quential criterion guarantees that g(xn) ---; M. By the algebra of limits for
sequences,

Thus,\;/ sequences in [D(f) nD(g)]-{xo} 3 Xn----) Xo, f(xn). g(xn) ----) LM.
By the sequential criterion, we conclude that lim (f(x)g(x)) = LM. •
x--+xo

Theorem 4.2.11 (g): If lim f(x) = L and j(x) :'.'.'. 0 for all x in some
X--+Xo
Nf,(xo), then X--+XQ lim .JJ(X5 =.JI,.

Proof. Suppose {xn} is a sequence in D(f)-{xo} 3 Xn---; xo. Since J(x) :'.'.'.
0 for all x in some Nf,(x 0 ), we know that ft exists as x---; xo. Since lim j(x) =
X--+Xo
L , the sequential criterion (Theorem 4.1.9) guarantees that f(xn)---; L. By the
algebra of limits for sequences, J f (xn) ---; .JL.

Thus, \;/ sequences in D(f) - { xo} 3 Xn ---; xo, J f (xn) ---; .JI,. By the
sequential criterion, this tells us that lim .JJ(X5 = .JI,. •
X--+Xo

LIMITS OF POLYNOMIALS AND RATIONAL FUNCTIONS

Definition 4.2.12 A polynomial (in one variable) is a function of the form

where ao, a 1 , · · · , an are (constant) real numbers.

Theorem 4.2.13 (Limits of Polynomials) For any polynomial p(x) and
any xo E IR, lim p(x) = p(xo).
X--+Xo

Proof. By the "algebra of limits of functions," Vk = 0, 1, 2, · · · , n,

(

k
lim xk = lim x ) since xk is a product. Thus, by Lemma 4.2.10, lim xk =
x--+xo x--+xo x--+xo
x~. Then, by the algebra of limits, Vk = 0, 1, 2, · · · , n,

lim akxk = ak lim xk

X--+Xo X--+Xo

= akx~.