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Limits of a Function

A function y = f (x) is said to tend to limit u as x tends to a, if the difference between the values of f (x) and u becomes smaller and smaller, as the difference between the values of x and a reduces. It is represented as

 

Example: Consider a function f(x) = 5x
 
x = 2 ⇒ f(x) = 5 x 2 = 10
 
From the above step, we can say that as x approaches the value 2, f(x) approaches the value 10.
 
 
From the above table, we can see that the difference between the values of f(x) and f(2) becomes smaller and smaller, when the difference between the values of x and 2 becomes smaller. i.e., the value of f(x) approaches 10 as the value of x approaches 2. This can be shown as Description: 87495.png
 
In the above table, you can see that there are two different cases of x approaching 2, i.e.x approaching 2 from a value lesser than 2 and x approaching 2 from a value greater than 2.
 
The limit of a function, when x approaches the value from a value lesser than it, is known as left hand limit. For the above table, the left hand limit is shown as Description: 87511.png
 
The limit of a function, when x approaches the value from a value greater than it, is known as right hand limit. For the above table, right hand limit is shown as Description: 87525.png

Rules on limits

Let where u and v are finite quantities:

 

Methods of calculating limits

Direct substitution method

Example
Description: 87673.png
Solution
Substituting x = 2 in the given function, we get
 
Description: 87732.png 

 
Example
Description: 87746.png
Solution
Substituting x = 3 in the given function, we get
Description: 87753.png 

 

Note: Any number of the form Description: 87944.png is known as an indeterminant form.


After substituting the value for the variable, if the final answer you get is an indeterminate form, then you can solve limits by the following methods:

Factorisation method

In this method, we factorize the functions under the limit into simpler functions and see if any terms can be cancelled. Then, we proceed with substituting the value for the variable and calculating the limit.
 

Example
Description: 87958.png
Solution
Let us try solving this using direct substitution method.
Substituting x = 1, we get
Description: 88016.png 
Description: 88023.png is not defined. Hence, we cannot solve the above problem by direct substitution.
Let us try factorizing the numerator and the denominator.
We get Description: 88029.png 
Description: 88037.png 
Substituting x = 1, we get Description: 88043.png

 

Example
Description: 88049.png
Solution
Let us try solving this using direct substitution method.
Substituting x = 2, we get
Description: 88056.png 
Description: 88062.png is not defined. Hence, we cannot solve the above problem by direct substitution.
Let us try factorizing the numerator and the denominator.
We get Description: 88087.pngDescription: 88098.png 
Substituting x = 2, we get
Description: 88124.png 

Solving by rationalization

We use this method of solving limits, when polynomials under the square-root sign are added or subtracted. First, we rationalize the function and then compute the limits by any of the methods given above.
 

Example
Description: 88189.png
Solution
If we try solving the above limit by direct substitution method, we will end up with an indeterminate form. Also, factorizing the functions will not help us as no terms would get cancelled. So, let us try to rationalize the function and then calculate its limit.
Multiplying the numerator and denominator of the function by Description: 88195.png
 
We get, 
 
 Description: 88223.png 
 
Now, substituting x = 0, we get
 
Description: 88234.png 

L’Hospital rule

L’Hospital rule uses derivatives to help evaluate limits involving indeterminate forms.  is an indeterminate form, then using L’Hospitals’ rule,  where f (x) and g(x) are the derivatives of f (x) and g(x).

 

Example
Description: 88309.png
Solution
If we try solving the above limit by direct substitution, we will get an indeterminate form. Using L’Hospital rule, we can solve by differentiating both numerator and denominator.
 
Description: 88321.png
 
Substituting x = 1, we get
 
Description: 88337.png 
which is again an indeterminate form. Applying L’Hospital rule again, we get
 
Description: 88348.png 
 
Substituting x = 1, we get
 
Description: 88385.png 
 
which is again an indeterminate form. Applying L’Hospital rule again, we get
 
Description: 88396.png 
 
Substituting x = 1, we get
 
Description: 88410.png 
 
Hence,
 
Description: 88416.png 

 

Note: As we saw in the Example 9.9, we need to go on applying L’Hospital rule until the end result is not an indeterminate form.

Some important standard limits

  • Description: 88422.png 

Example
Description: 88618.png
Solution
Given
Description: 88629.png 
Description: 88640.png  
Example
Description: 88653.png
Solution
If we substitute x = 0 in the given limit, we get the indeterminate form 1
Description: 88724.png
 

Example
Description: 88740.png
Solution
Multiplying and dividing the function by 2, we get,
Description: 88752.png 
Example
Description: 88794.png
Solution
This is of the form Description: 88801.png which is equal to Description: 88807.png
Description: 88815.png 
Example
Description: 88854.png
Solution
Multiplying and dividing the function within the limits by 2, we get,
Description: 88867.png  
Example
Description: 88898.png
Solution
Let us try to convert the given limit into one of the standard forms.
Description: 88911.png  
  • Description: 88949.png
Example
Description: 88955.png
Solution
Let us try to convert the given limit into one of the standard forms.
Description: 88983.png 

 

Example
Description: 88967.png
Solution
Let us try to convert the given limit into one of the standard forms.
Description: 89004.png 
Description: 89014.png 

 

Note:

 

Example
Description: 89229.png
Solution
Description: 89240.png
Now this is a standard limit and we know that Description: 89254.png 

 

Example
Description: 89261.png
Solution
Applying L Hospital’s rule
Description: 89308.png 





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