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Limits


We shall see the meaning and definition of limits, learn some of their properties and derive the formula for a few standard limits.

Limit of a function:
The idea of limit of a function is related to the idea of nearness or closeness of the function to a particular value. It comes into play in situatious where one quantity depends on another varying quantity and we have to know the behaviour of the first when the second is very close to a fixed given value.

Let us look at some examples:

Example: 1
Let us consider a real valued function
See the tables I and II given below :

Table I

x

1

1.2

1.4

1.6

1.8

1.9

1.99

1.999

4

4.2

4.4

4.6

4.8

4.9

4.99

4.999


Table II

x

3

2.8

2.5

2.2

2.1

2.01

2.001

2.00001

6

5.8

5.5

5.2

5.1

5.01

5.001

5.0001


These tables give values of as gets closer to 2 through values less than 2 (in table I ) and greater than 2 (in tables II)
From these tables, we can see that as  approaches 2, approaches 5; the nearer the value of chosen to 2, the nearer is to 5. Then 5 is the value of as approaches 2. We call such a value the limit of as tends to 2 and is denoted by,


Here the value of the limit coincides with the value of .

Note: 1
There is difference between and . means that get closer and closer to 2, but never becomes equal to 2, whereas, means, takes the value 2.

Note: 2
Let us now consider the real valued function given by This function is not defined at since division by zero is not defined.

Example: 2
Consider the following tables, which give values of as approaches 2 through values less than 2 and through values greater than 2.

Table I

1

1.5

1.9

1.99

1.999

3

3.5

3.9

3.99

3.999


Table II

3

2.5

2.1

2.01

2.001

5

4.5

4.1

4.01

4.001


We see that approaches 4 as approaches 2. Hence

This can be done algebracially thus: 
This cancellation is possible only because .
Hence

Example: 3
Now take another example. Let Here is not defined. Let us try completing tables, as approaches 0.

Table I
1 0.9 0.5 0.3 0.1 0.001
1 1.1111 2 3.3333 10 1000
Here values are approaching 0 through values greater than 0.

Table II

0.5

0.1

0.001

0.00001

2

10

1000

100000

We see that does not approach any fixed number as approaches 0 .
In this case does not exist.
[Mathematically, this limit is ]

Conclusion

  1. There are functions whose limit exist and they are the same value of the value of function at the given point.

    These limits can be obtained by direct substitution.
  2. There are functions whose limit exist but cannot be determined by direct substitution.
  3. There are functions whose limit may not exist.
    These examples lead us to the definition of limit.

Definition: Limit of a function


Let be a function of a real variable . Let be two fixed numbers. If approaches as approaches , then we say that is the limit of the function as tends to .

 

Left hand and Right hand limits


While defining the limit of a function as  xtends to , we consider values of when is very close to . The values of may be greater or less than [Refer to tables constuencted for examples (1) (2) (3) in the previous pages]. If we restict less than , then we say that from below or left and represent it as Similarly, if takes only values greater than , then is said to tend to from above or right and is denoted by .


Note: It is important that for the existence of , it is necessary that

Fundamental Properties of Limits


a constant for all values of , then
If for all values then
for being a constant.





Some standard limits


For any rational index

Eventhrough the result is true for rational indices, we shall prove the result for only a positive integer .

Proof:
Dividing



Proof:

We take . This function is defined for all values of , other than .
Since  the value does not change.
Therefore it is enough if we find the limit asthrough positive values of .
Let and be two radii of a circle, with centre.
Let unit. and are tangents from an external point to the circle. Let be the angle subtended by the arc at the centre. If is measured in radians, then

Since is right angled at (being a tangent which is perpendicular to radius )

So we get, Arc AB =
           Chord AB = 2 sin x.
                    PA + PB = 2 tan x.
Obviously, 
Dividing through out by we get.



Note:
  1. For this result to be, the angle must be measured in radians.
  2. The significance of this result is that as the angle gets smaller and smaller (nearing zero), the value of is nearly equal to (in radians).
The following is a table to illustrate the property stated above [Angles are in radians] Remember
Sin 0.1 Sin 0.2 Sin 0.3
0.09983341665 0.1986693308 0.2955202067
Sin 0.15 Sin 0.25 Sin 1
0.1494381325 0.2474039593 0.8414709848
Sin 2 Sin 0.5 Sin 1.5
0.9092974268 0.4794255386 0.9974949866
Table clearly shows that as the difference of the angle from is more, the value of the sine of the angle is also far away from the value of the angle (marked values).

Let us consider
Consider
Using property (6) of limits, we get.

                           
    
Hence the result

Example: 1


Solution:
Method (1)

(or)

Method (2):
Use
Here


Example: 2
Find the right hand and the left hand limits of the function at


Solution:
When


When

Therefore left limit and right limit both exist; but they are not equal.

Example: 3
Find , if it exists.

Solution:
Right limit:


Left limit:

Left limit ≠ Right limit
the limit does not exist.

Example: 4
Evaluate

Solution:


Example: 5
Evaluate

Solution:

As



Example: 6
Compute

Solution:
Factorising the numerator, we get
Factorising the denominator, we get


Example: 7
Find

Solution:
Put



Example: 8
Evaluate

Solution:
As
Divide each term in the numerator and denominator by , we get



Example: 9
Evaluate the left and right limits of does the limit of exist? Justify your answer.
Find the positive integer such that

Solution:



Example: 10
If  then prove that


Solution:
Given:


Working Rule for calculating limits


Let us take

Step 1: Substitute in the given function of the result is a finite quantity (including 0) that is the required limit.
Step 2: Substituting may give rise to an indeterminate quantity like In such cases limit (if it exists) can be found out by removing some common factor in numerator and denominator (which becames zero when ) or bringing the question to some standard form whose limits are known.




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