Module 2
: Limits and Continuity of Functions
Lecture
4
:
Limit at a point
(ii)
Let
Then,
Thus,
does not exist.
4.1 .13
Example :
Let
. To analyze
, consider
.
Then,
and
.
However, if we consider
and for every
.
Hence,
. Thus, though both
and
converge to
, but
and converge to different limits.
Hence, limit does not exist, by the previous theorem.
4.1 .14
Note :
Theorems similar to that of theorem 2.1.3 hold for left-hand and right-hand limits.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
does not exist. 
. To analyze 
, consider 
.
and 
. 
and for every 
. 
. Thus, though both
and
converge to 
, but 
and converge to different limits.
