Module 2
: Limits and Continuity of Functions
Lecture
4
:
Limit at a point
Click here to see an interactive visualization :
Applet 2.4
Before proceeding further, we show that the existence of limit is equivalent to the existence of the
limit.
4.1 .8
Theorem :
For a function
, the
limit exists at a point
if and only if
, i.e., for every sequence
with
for all
, we have
.
4.1 .9
Note :
(i)
depends on the values of
at points near
c
. The function
may or may not be defined at
c.
Even if
is defined at
c
,
may or may not exist. Even if
exist, it need not be equal to
.
(ii)
To find
, one has to make a guess and then prove it.
Let us note that,
means that for a given
, there exists
such that for all
,
implies
.
Equivalently,
there exists
such that
implies
and
implies
.
This motivates our next definition.
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limit. 
, the 
if and only if 
, i.e., for every sequence 
with 
for all 
, we have 
. 
depends on the values of 
at points near
.
means that for a given 
, there exists 
, 
implies 
.
there exists 
implies 
and 
implies 
