What is Limit in Calculus? Introduction

                                                                         Part -I

WHAT IS LIMITS?                                     

 To know what is limit, let us consider on following function:

CASE 1:-

f(x) = x² – 4

           x – 2

Obviously, it is clear that this function is defined for all x except x =2 as it considers the form 0/0 (known as an indeterminate form) at  x = 2,

However, if x ≠ 0, then

F(x) = (x -2)(x +2)  = x + 2                                   [ x² – 4 = ( x – 2)² = ( x -2)(x + 2)]  

                x – 2

If we put  different values of x less than 2 and more than 2, then we will get the following result which is shown in following  tables:-

Table (a):

X	1.4	1.5	1.6	1.7	1.8	1.9	2.00	2.01	2.1	2.2	2.3	2.4	2.5	2.6	2.7
F(x)	3.4	3.5	3.6	3.7	3.8	3.9	0/0	4.01	4.1	4.2	4.3	4.4	4.5	4.6	4.7

Graph 1,

It is evident from table (a) and graph 1, of f(x) that as x increases and reaches closer to 2 from left hand side of 2, the values of f(x) increase and approaches to 4.  We can define it as:-

When x tends to 2 from lest hand side, the function f(x) increase and come closer to 4.

If we use the notation x →  ‘to denote’ x tends to 2 from left hand side’.

Or,  we can say,

x→2 ^– , f(x) →4 or limit f(x) =4

                               x →2 ^–  

or Left hand limit of f(x) at x =2 is 4,

Thus, limit f(x) = 4

           x→2 ^–  

Means that as x tends to 2 from left hand side , f(x) is tending to 4.

From the table (a) and graph 1, which showed f(x)? We observed that as x decreases and approaches to 2 from right hand side 2 from right hand side 2, the values of f(x) decreases and come closer to 4 

or we can say:

When x tends to 2 from its right hand side, the function f(x) approaches to the limit 4.

Using the rotation x → 2⁺ to denote x tends to 2 from right hand side, the above statement can be re-started as

x → 2⁺, f(x) →4

or,  limit f(x) = 4

       x → 2⁺

or, right hand limit of  f(x) at x = 2 is 4.

Thus, limit f(x) = 4

       x → 2⁺

Means that as x tends to 2 from right hand side , f(x) is tending to 4.

 

On the basis of above discussion of function f(x), we have

f(x) = x² – 4

           x – 2

(i) f(x) = 4

(ii). limit f(x) = 4

        x → 2^–

(iii) limit f(x) = limit f(x)

        x → 2^–        x → 2⁺

(iv) f(2) does not exist, i.e., f (x) is not defined at x = 2.

CASE 2. Let us consider on following function

F(x) =|x – 3|

            x – 3          

This function is defined for all x except x =3, as it supposes the form 0/0 (an indeterminate form) at x =3, 

If we put  different values of x less than 3 and more than 3, then we will get the following result which is shown in following  tables:-

Table (b):- 

X	2.4	2.5	2.6	2.7	2.8	2.9	3	3.01	3.1	3.2	3.3	3.4	3.5	3.6	3.7
F(x)	– 1	– 1	– 1	– 1	– 1	– 1	0/0	1	1	1	1	1	1	1	1

 Graph no:-2.

It is evident from this table and graph of f(x) that as x à3 from lfet hand side th values of f(x) are everywhere  – 1

i.e., limit f(x) =  – 1

       x →3 ^–

or, Left hand limit of f(x) at x = 3 is  –1

We also observed that at every point on the right hand side of 3, the function has same value +1,

Therefore,

Limit f(x) = +1

 x →3⁺

CASE 3:-Let us now consider the function f(x)=          1          ., x ≠4

                                                                                        x – 1

Here, the function is undefined at x = 4 as f(x), suppose the form 1/0. In this case it is evident from the graph shown in graph 4. That as x approaches to 4 from the left hand side, f(x) decreases to   – ∞ .

i.e, limit f(x) = – ∞

x→ 4⁺

and f(x) increases to + ∞ as x approaches to 4 from the right, i.e.

limit f(x) = + ∞

x→+ ∞

So, we say that  limit f(x) = – ∞   and Limit f(x) = +1 both do not exist.

                                      x →3⁺                          x→ 4⁺

Now, we reach at the conclusion from the above discussion that we can approach to a given number ‘a’ (say) on the real line either from its left hand side by increasing number which are less than ‘a’ or from right hand side by decreasing numbers which are greater than ‘a”.

Hence, there are two types of limit, (i) left hand limit and (ii) right had limit. We also find that for some functions at a given point ‘a’ left hand and right hand limits are equal wherever for some functions these two limits are not equal and even sometimes either left hand limit or right hand limit or both do not exist.

If limit f(x) = limit f(x)

   x→ a ^–      x →a⁺

i.e., (LHL at x = a) =(RHL at x = a)

Then we can say that limit f(x) exits.

                                       x→a

Otherwise, limit f(x) does not exist.

                     x→a