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 
Table II
âˆ’ 0.5 
âˆ’ 0.1 
âˆ’ 0.001 
âˆ’ 0.00001 

âˆ’ 2 
âˆ’ 10 
âˆ’ 1000 
âˆ’ 100000 
In this case does not exist.
[Mathematically, this limit is ]
Conclusion
 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.  There are functions whose limit exist but cannot be determined by direct substitution.
 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:
 For this result to be, the angle must be measured in radians.
 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).
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 
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:
âˆ´ Left limit â‰ Right limit
âˆ´ 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.