1549901369-Elements_of_Real_Analysis__Denlinger_

62 Chapter 2 • Sequences

--(a)

• (c)

__.....(e)

--(g)

./ (i)

lim --; = 0

n--tex> n

7n

lim --=0

n-+= n^2 + 3

lim ~ =3

n-+= n + 4

lim 3n+4 = ~

n-+= 7n - 1 7

lim Sn = 0

n-+= 11 + n^2

n^2 - 2

lim ---= 1

n-+= n^2 + n

n - 2n^2 2

/ (m) lim =--

n-+= 3n^2 + 1 3

n^2 +3n

./ ( o) lim 2 = -1

n-+= 10 - n

. 3
(b) hm --= 0
n-+= n + 4

11

(d) lim -- 2 = 0

n-+= 1 -n

( f) lim^2 n -^5 = 2

n-+= n - 6

2n 2

(h) lim --= --

n-+= 1 - 5n 5

(j) lim n = 0

n-+= 1 + 8n^2

r 8n

2

+ 3

(1) n~~ 5n^2 - 2n

8

• 5

2n^2 -n

(n) lim = 2

n-+<Xl n^2 - 5n - 7

(p) lim n 2 + 6n = 0

n-+<Xl n3 - 5n + 1

3. Use the methods of this section to prove each of the limit statements
   (a)- (p) given in Exercise 2 above.

2.2 Algebra of Limits

In this section we establish some basic rules that allow us to evaluate limits
algebraically, without resorting to c-n 0 arguments.

Theorem 2.2.1 (Absolute Value and Limits) Suppose {xn} is a sequence.
Then

(a) Xn-+ 0 {:} lxnl-+ O;

(b) Xn-+ L {:} lxn - LI-+ O;

(c) Xn-+ L =? lxnl-+ ILi. (Note: we do not claim "{:}. ")

Proof. Exercise 1. •

Definition 2.2.2 A sequence { Xn} is called a constant sequence if 3 c E JR 3
'<In E N,xn = c.