Module 1
:
Real Numbers, Functions and Sequences
Lectur
e
3
:
Monotone Sequence and Limit theorem [ Section 3.2 : Limit Theorems on Sequences ]
Hence,
implying that
.
Thus
. For x = 1, clearly
for every
, and hence
Practice Exercises 3.2 : Limit Theorems
(1)
Show that the following limits exists and find them:
(i)
.
(ii)
.
(iii)
.
(iv)
.
(v)
.
(vi)
.
(2)
If
find
.
(3)
If
show that there exists
such that
(4)
If
and
show that
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
implying that 
. 
. For x = 1, clearly 
for every 
, and hence 
.
.
.
.
.
.
find 
.
show that there exists 
such that 
and 
show that 
