4.4 *Infinity in Limits 215Proof. (a) Exercise 11.
(b) Exercise 12. •ALWAYS REMEMBER that +oo and -oo are not real numbers.
We should not expect them to obey all the rules of the algebra of real num-
bers. They are merely convenient symbols, which seem to obey some common
algebraic rules. They are intended for use only in connection with limits.
EXERCISE SET 4.4-A- Use Definition 4.4.1 to prove the following limit statements:
1 -1
(a) xlim --+O -X 2 = +oo (b) x--+1 lim ( X - 1 ) 4 = -oo
. 1
(c) X--+-1 hm ( X + 1 ) 2 = +oo
1-x
(e) X--+2 lim ( X - 2 ) 2 = -oo
(d) lim ( x ) 2 = +oo
x--+1 X - 1(f) lim ( x + \ 2 = -oo
x--+-3 X + 3- Modify Theorem 4.4.3 to yield a correct theorem about lim f(x) = -oo.
X--+XQ - Define each of the following:
(a) lim f(x) = +oo
x--+xQ
(b) lim_ f(x) = -oo
(c) lim f(x) = +oo
x-.xt(d) lim f(x) = -oo
X--+Xt- Modify Theorem 4.4.3 to yield correct theorems about lim f(x) = +oo,
X--+X 0
lim f(x) = -oo, lim f(x) = +oo, and lim f(x) = -oo.
x--+xQ x-.xci x--.xci
In Exercises 8-20, the generic symbolic statements limf(x) = +oo and
limf(x) = -oo will be understood to cover all three possibilities: lim,
lim , or lim.
x--+xci x--+xo- Revise Theorem 4.1.9 to a correct theorem about limf(x) = +oo and a
correct theorem about lim f ( x) = -oo. - Revise Corollary 4.1.10 to a correct theorem about limf(x) = +oo and a
correct theorem about limf(x) = -oo. - Prove Theorem 4.4.6.
- Prove Theorem 4.4.8 (b).
- Prove Theorem 4.4.8 (c).