2.4 Divergence to Infinity 83Thus, we want to find an no E N 3
3n- 2
n 2:: no :::;, n 2:: 24 and -
6
100; i.e.,
n 2:: no :::;, n 2:: 24 and 3n - 2 > 600; i.e.,
n 2:: no :::;, n 2:: 24 and 3n > 602.This will be satisfied if n ;::: 201. Thus, take no = 201.
3n^2 - 2n
(b) We want an n 0 EN 3 n > n 0 ::::? > M. As shown above,- 5n+ 23
3n^2 - 2n 3n - 2
5n + 23 > -6-, if n;::: 24.
Thus, we want to find an no E N 3
3n- 2
n 2:: no:::;, n 2:: 24 and -
6
M; i.e.,
n 2:: n 0 :::;, n 2:: 24 and 3n - 2 > 6M; i.e.,
n 2:: no:::;, n 2:: 24 and 3n > 6M + 2.
This will be satisfied if n > 24 an d n > 6M +2 Th k
3. us, we ta e no >
{
max 24, 6M + 2}
3
. D
. (3n
2
Example 2.4.3 Prove that hm - 2n)
23= +oo.
n--->oo 5n +
Solution: Let M > 0. By the Archimedean property, 3 no E N 3 no >{
max 24, 6M +2}
3
. Then,
6M+2
n 2:: no :::;, n >
3
and n 2:: 24:::;, 3n > 6M + 2 and n ;::: 24
3n- 2
:::;, -
6
M and n ;::: 24
3n^2 - 2n
:::;,
6n > M and n ;::: 24
* 3n^2 - 2n 3n^2 - 2n M
5n + 23 > 6n >