1549901369-Elements_of_Real_Analysis__Denlinger_

2.1 Basic Concepts: Convergence and Limits 57

I

Now, n > 3 ::::}^23 I^23

9

n _

21

=

9

n _

21

. Thus,

n ::'.". no ::::} I 23 I < c

9n - 21

I

6n + 9 - 6n + 141

::::} 3(3n - 7) < c

I

3(2n + 3) - 2 (3n - 7) I

::::} 3(3n - 7) < c

::::} I 2n + 3 _ ~I < E:.

3n- 7 3

That is,. :lno EN 3 n ::'.".no::::} I ---2n +^3 - -^21 < c.

3n- 7 3

Therefore, hm. (2n ---+ 3) = -^2 by Defimt10n .. 2.1.l.

n->oo 3n - 7 3

D

The type of reasoning used in Examples 2.1.5 and 2.1.6 is so important
in analysis that we shall give another pair of examples. Compare these new
examples with 2.1.5 and 2.1.6 to see where they are similar. To facilitate the
comparison, we will ask the same three questions we asked in Example 2.1.5,
and ask for a "proof" just as we did in Example 2.1.6.

Example 2.1.7 Consider the limit statement lim (
3
n:-

4

n) = 3.

n->oo n + 5

( ) Af h d h 3n

(^2) - 4n. h" d.
a ter ow many terms are we guarantee t at 2 is wit m a is-
n +5
tance of .0 1 of 3?
(b) After how many terms are we guaranteed that the nth term of this se-
quence is an accurate approximation of the limit, to within 3 decimal
places?
(c) For arbitrary c > 0, after how many terms are we guaranteed that
3n2 - 4n.. h. d. f 3?
n 2 +
5
is wit m a istance c o.