1549901369-Elements_of_Real_Analysis__Denlinger_

212 Chapter 4 • Limits of Functions

x-2

Example 4.4. 7 Investigate lim --.

x-+l X - 1

x-2

Solution: In Example 4.4.5, we showed that lim --= +oo and

x-+1-x-l

lim x -

2

= -oo. Theorem 4.4.6 then tells us that lim x -

2

is neither +oo

x-+l+X-1 x-+lX-1

nor -oo. The most we can say about lim x -

2

is that it does not exist. D

x-+l X - 1

ALGEBRA OF INFINITE LIMITS

Theorem 4.4.8 Suppose lim f(x) +oo, lim g(x) +oo, lim h(x)

x--+xo X-+Xo X--+Xo

-oo, and lim k(x) = -oo. Then

X--+Xo

(a) lim (f(x) + g(x)) = +oo;

X--t-Xo

(b) lim (h(x) + k(x)) = -oo;

X--+Xo

(c) lim (f(x)g(x)) = +oo;

x--+xo

(d) lim (h(x)k(x)) = +oo;

X--+Xo

(e) lim (f(x)h(x)) = -oo.

X--+Xo

Proof. (a) Suppose lim f(x) = +oo and lim g(x) = +oo. Let M > 0.

X--+Xo X--+Xo

Since X--+Xo lim f(x) = +oo, 381 > 0 3 0 < Ix - xol < 81 =} f(x) > M. Since

X--+Xo lim g(x) = +oo,^382 >^0 3 0 < Ix - xol <^82 =} g(x) > M. Let^8 =
min{81,82}. Then,

0 <Ix - xol < 8 =} f(x) >Mand g(x) > M

=} f(x) + g(x) > 2M > M.

Therefore, by Definition 4.4.1, lim (f(x) + g(x)) = +oo.

(b) Exercise 8.

( c) Exercise 9.

(d) Exercise 10.

X--+Xo