This document contains a tutorial on limits of sequences. It provides examples of calculating the limit of several sequences as n approaches infinity. It also gives examples of finding the limit supremum and limit infirmum of sequences. The document concludes by wishing the reader good luck on their exercises and reminds them to remember Allah.

1. In the name of Allah
the most gracious, the most merciful

2. TUTORIAL 8

3. 6. Find the limit of the following sequences.
𝑎) 𝑈 𝑛 = 𝑛 + 1 − 𝑛
lim
𝑛→∞
𝑈 𝑛 = lim
𝑛→∞
𝑛 + 1 − 𝑛
lim
𝑛→∞
𝑈 𝑛 = lim
𝑛→∞
𝑛 1 +
1
𝑛
− 𝑛
lim
𝑛→∞
𝑈 𝑛 = lim
𝑛→∞
𝑛 1 +
1
𝑛
− 1
𝑛 → ∞,
1
𝑛
→0
lim
𝑛→∞
𝑈 𝑛 = ∞ 1 + 0 − 1
lim
𝑛→∞
𝑈 𝑛 = ∞ 0 = 0
𝑏) 𝑈 𝑛 = 2 𝑛 + 3 𝑛 1/𝑛
lim
𝑛→∞
𝑈 𝑛 = lim
𝑛→∞
2 𝑛 + 3 𝑛𝑛
lim
𝑛→∞
𝑈 𝑛 = lim
𝑛→∞
3 𝑛
2 𝑛
3 𝑛
+ 1
𝑛
lim
𝑛→∞
𝑈 𝑛 = lim
𝑛→∞
3.
2 𝑛
3 𝑛 + 1
𝑛
lim
𝑛→∞
𝑈 𝑛 = 3 lim
𝑛→∞
2
3
𝑛
+ 1
𝑛
𝑛 → ∞,
2 𝑛
3 𝑛 →0
lim
𝑛→∞
𝑈 𝑛 = 3 1 = 3

4. 6. Find the limit of the following sequences.
𝑐) 𝑈 𝑛 = 𝑛2 + 𝑛 − 𝑛
lim
𝑛→∞
𝑈 𝑛 = lim
𝑛→∞
𝑛2 1 +
1
𝑛
− 𝑛
lim
𝑛→∞
𝑈 𝑛 = lim
𝑛→∞
𝑛 1 +
1
𝑛
− 1
𝑛 → ∞,
1
𝑛
→0
lim
𝑛→∞
𝑈 𝑛 = ∞ 1 + 0 − 1
lim
𝑛→∞
𝑈 𝑛 = ∞ 0 = 0
𝑑) 𝑈1 = 2 , 𝑈 𝑛+1 = 2 + 𝑈 𝑛
𝐿𝑒𝑡 lim
𝑛→∞
𝑈 𝑛 = 𝐼 = lim
𝑛→∞
𝑈 𝑛−1 = lim
𝑛→∞
𝑈 𝑛+1
lim
𝑛→∞
𝑈 𝑛 = lim
𝑛→∞
2 + 𝑈 𝑛−1 = 𝐼
2 + 𝐼 = 𝐼
2 + 𝐼 = 𝐼2
𝐼2
− 2 − 𝐼 = 0
𝐼 − 2 𝐼 + 1 = 0
𝑆𝑜, 𝑡𝑎𝑘𝑒 𝐼 = 2
lim
𝑛→∞
𝑈 𝑛 = 2

5. 6. Find the limit supremum and infremum of the following sequences.
𝑎) 𝑈 𝑛 = −1 𝑛+1
= 1, −1,1, −1,1, −1, …
𝑀1 = 1, 𝑀2 = 1, 𝑀3 = 1, …
𝑚1 = −1, 𝑚2 = −1, 𝑚3 = −1, …
lim sup 𝑈 𝑛 = lim 𝑀 𝑛 = 1
lim inf 𝑈 𝑛 = lim 𝑚 𝑛 = −1
𝑏) 𝑈 𝑛 = 1 +
−1 𝑛
𝑛2 = 0,
5
4
,
8
9
,
17
16
,
24
25
, …
𝑀1 =
5
4
, 𝑀2 =
5
4
, 𝑀3 =
17
16
, …
𝑚1 = 0, 𝑚2 = 0, 𝑚3 = 0, …
lim sup 𝑈 𝑛 = lim 𝑀 𝑛 = 1
lim inf 𝑈 𝑛 = lim 𝑚 𝑛 = 0
𝑐) 𝑈 𝑛 = −1 𝑛+1
𝑐𝑜𝑠
𝑛𝜋
4
=
1
2
, 0, −
1
2
, 1,
1
2
, 0, …
𝑀1 =
1
2
, 𝑀2 = 1, 𝑀3 = 1, …
𝑚1 = −
1
2
, 𝑚2 = −1, 𝑚3 = −1, …
lim sup 𝑈 𝑛 = lim 𝑀 𝑛 = 1
lim inf 𝑈 𝑛 = lim 𝑚 𝑛 = −1
𝑑) 𝑈 𝑛 =
−1 𝑛+1
𝑛
+
−1 𝑛
𝑛2
= 0,
−1
4
,
2
9
,
−3
16
,
4
25
, …
Exercise

6. • Question 8 and 9 have been discussed in your
lecture.
• Thank you, Good luck.
• Banyakkan latihan. Moga Berjaya. Ingat Allah
selalu.