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Limit of a function Definition of a function : Let A and B be
two sets and f be a mapping from A into B. Then f
is said to be a function from A into B if for every element x in
A an unique element y in B is
assigned by f and it is denoted by :f A B . The element y in B
is called the image of x under f.
The sets A and B are called Domain and Co-domain
respectively.
The set of all images in B is called the Range of the function f
and is denoted by R(f).
If A and B are subsets of the real line R, then f is said to be
a real valued function of a real
variable.
Limit of a function at a point a A :-
Let :f A B be a real valued function of a real variable. Let a R
and let A contains all
points in some open interval ( , )a h a h except possibly the
point a itself. Then
Definition of limit : A real number L is called the limit of the
function at the point a R if
for any given 0 there exists 0 such that | (x) L |f whenever | x
|a .
It can also be written as (x)L f L whenever a x a .
We denote this as (x)limx a
f L
.
From the definition, it is clear that as x approaches a, f(x)
approaches the value L. Note that the
point a need not be in the domain A.
Example : Let :(0,1) (1,2)f R as
3 (0,1)
(x)4 x (1,2)x if x
fif
Here the point 1 does not belong to the domain of f. Now it can
be easily seen that
1(x) 4lim
xf
; because for 0 , 4 (x) 4f whenever 1 1x .
Example : Using the definition of limit, prove that
21
3 4 2limx
x x
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In order to prove that 1
(x) 2limx
f
, let 0 be given. Here 2(x) 3x 4 and L 2f x .
Now to prove that
| (x) L | | (x 1)(x 2) |f whenever | 1 |x (1)
for some 0 depending on .
If there exists such a 0 , then we get 1 1x
1 2 1
| 2 | 1| 1 | | x 2 | (1 ).
xx
x
Now if is chosen such that (1 ). , then for such choice of ,
condition (1) is satisfied.
i.e 1 1 42
Example : Prove that 1
1coslimx x
does not exist.
If we assume that there exists a limit L, then we arrive at a
contradiction.
Let 0 be given. Then by assumption, there exists a 0 such that
1| cos L |x
whenever
| 0 |x .
Now we observe that if 1 12 , cos 02
nx x
and if 1 12 , cos 1n
x x .
(4 1), | cos L | | |2
nFor all n L (2)
and | cos(2 1) L | | 1 |2
n L (3)
If we choose 14
, then there exists a 1N
for some positive integer N. Now we can find
integers n greater than N such that when 1xn
, | |x and 1 1| | | L 1 |4 4
L and ,
Which is not possible for any L.
Hence the limit does not exist.
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Right and left limit of a function at a point:
Right limit: L R is said to be the right limit of the function f
at the point a R , if for any
given 0 , there exists a 0 such that | (x) L |f whenever a x a
.
The above definition is denoted by (x) Llimx a
f
. This means that when x approaches the value
a from the right side, f(x) approaches the value L.
left limit: M R is said to be the left limit of the function f
at a , if for any given 0 , there
exists a 0 such that | (x) |f M whenever a x a .
We denote this by (x)limx a
f M
. This means that when x approaches the value a from the
left
side, f(x) approaches the value M.
If the left and right limits at a point are equal, then we say
that the limit exists at that point. i.e. ,
if (x) (x) Llim limx a x a
f f
, then (x) Llimx a
f
.
Ex:- Let :[0,2]f R be defined as
1 : [0,1)
(x)2 : x [1, 2]
xf
Then it can be easily seen that 1 1
(x) 1 (x) 2lim limx x
f and f
,
Hence the limit does not exist at 1x .
