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Introduction to Limits

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Introduction to Limits - Lesson Summary

There are some situations where we wish to find a point or situation where we are interested in a particular outcome.

Consider the sequence  1 2   ,  2 3   , 3 4   , 4 5   ,....  

To obtain 1, the sequence needs to be extended infinitely. 

Another example: In an ‘n’-sided polygon inscribed in a circle, finding the value of ‘n’ for which the area of the polygon becomes equal to the area of the circle.
 
Start with the three-sided polygon, the triangle.
 
The area of the triangle is not equal to the area of the circle.
If the number of sides of the polygon is increased to 12, we can see the distinction between the polygon and the circle.
 


 
If the number of sides of the polygon is increased to 17, we can observe that the area of the polygon is almost equal to the circle, but not actually equal.
 


 
As we keep on increasing the number of sides of the polygon, the area almost becomes equal to the area of the circle.
 
This can be summarised as ‘As the number of sides, n, tends to infinity, the area of the polygon equals the area of the circle.’
 
This can be mathematically represented as n → ∞, A p → A c

A p = Area of the polygon

A c = Area of the circle
 
Or concisely  lim n → ∞ A p = A c
 
When the number of the sides of the polygon becomes infinity the area of the polygon will become equal to the area of the circle.  
 
 Suppose a function, f(x) = x 2  which is a parabola.
 
Finding the value of the function as x approaches an arbitrary point, a.
 
The mathematical representation of this idea is represented as → a, f(x) → l.

As the value of x approaches a, the value of the function approaches l.

This is symbolically defined as lim x → a f(x) = l .
 
This is read as—: Limit x tends to a ... f of x is equal to l’.
 
Ex: To find what value of the function approaches as x approaches two.
 

 
It can be observed that the value of the function approaches 4.                
 
Symbolically, it is stated as lim x → 2 x 2 = 4 .

This means that as x approaches 2, the value of the function approaches 4.
                                                     
                            x → 2, f(x) → 4
 
Left hand limit and Right hand limit
 
Consider the function,  f(x) = x 2.
 
When x tends to point a, ‘a’ can be approached in two ways.

 ‘a’ can be approached either from the right hand side or from the left hand side.

When the limit is found from the left hand side, the limit is called as the left hand limit.


Similar is the case with the right hand limit.


Right hand limit and the left hand limit are shown as

Right hand limit : lim x → a + f(x)

Left hand limit : lim x → a – f(x)
 
The superscripts on the top indicate whether we are finding the right hand limit or the left hand limit.
For the function, x 2, the left hand limit is lim x → a – f(x)  = lim x → a - x 2 .
The value of the left hand limit is equal to 4.

       lim x → a - x 2   = 4
Similarly, the right hand limit is lim x → a + f(x)  = lim x → a + x 2 .
The value of the right hand limit is 4.
lim x → a + x 2   = 4 .
Thus, the left hand limit is equal to the right hand limit.

         
lim x → a + x 2   = 4  = lim x → a - x 2  

So, the limit of the function exist and is equal to 4 at x is equal to 2.
 
Consider the graph of the function, f ( x ) = 4x + 1 when x ≤ 0 x + 1 when x > 0
Evaluate the limit of the function at x equal to 1. For the limit to exist, we have to verify whether the left hand and the right hand limits coincide with each other.

Check : lim x → 1 - f(x)    = lim x → 1 + f(x)   

Left hand limit of the function:
                    lim x → 1 - f(x)    =  lim x → 1  (4x+1)    [f(x) = 4x + 1 when x ≤ 1]
The limit of the function is 5.
lim x → 1  (4x+1)  = 5
Right hand limit of the function:
                  lim x → 1 + f(x)   = lim x → 1 (x+1)   
                   lim x → 1 (x+1)  = 2
                    lim x → 1 - f(x)   =   lim x → 1 + f(x)  

Therefore, the limit of the function does not exist at x = 1.
Ambiguity exists. Therefore, the limit of this function at x = 1 does not exist.

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