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.