1549901369-Elements_of_Real_Analysis__Denlinger_

218 Chapter 4 • Limits of Functions

Theorem 4.4.18 (Fundamental Limits)

(a) \:In EN, lim xn = +oo;

x-.+oo

(b) \:In EN, ~f n is even, then lim xn = +oo;

X--+-00

(c) \:In EN, if n is odd, then lim xn = -oo.

x--+-oo

Proof. (a) Let n be a fixed natural number, and M > 0. Choose

N = M + 1. Then

x > N ::::} x > 1 and x > M

::::} xn > x and x > M

::::} xn >M.

Therefore, by Definition 4.4.12, lim xn = +oo.

x-.+oo

(b) Let n be a fixed even natural number, and M > 0. Then :3 k E N 3

n = 2k. Choose N = M + 1. Then

x < -N::::} -x > N

::::} lxl = -x > N > 1 and lxl = -x > M

::::} x2k = lxl2k > lxl > M

* x2k > M

::::} xn>M.

Therefore, by Definition 4.4.15, X--+-OO lim xn = +oo.

(c) Exercise 5. •

The following theorem shows relationships between lim f(x) and lim f (~)
x->O x->oo X
that are often useful.

Theorem 4.4.19 (a) lim f(x) = L (finite) if and only if lim f (~) = L;

x-.o+ x->+oo x

(b) lim f(x) = L (finite) if and only if lim f (~) = L.

X->O- X->-00 X