## Limits

Limits is an important topic in calculus and analysis.

### Introduction :

While calculating tangent slope problem, or average rate of change problem , we have to find how the  function f(x)  is behaving when x= α.

Some times we can not work out directly, but we can see where the function approaches.

For example, let us take the function

f(x) =  (x²-1)/x-1

When we find the value of the function when x = 1, we will get the value as in the form 0/0.

So let us find the value of the function as it approaches to 1.

Let us see the values of the function when x approaches 1 from the left.

Let us see when the values approaches 1 from the right.

From both the tables we came to know that the value of f(x) approaches to when x approaches 1.  In this type of situation mathematicians used the word limit.

We can say that the limit of the above function as x approaches 1 is 2.

Definition:

Limit 'L' is the value that the function f(x) or a sequence approaches as x approaches  α.

We can write this mathematically as

Lim  f(x)  = L

Graphical approach :

The following graph shows when x approaches from both left and right the value of function f(x) approaches 1.

The graph shows that

limₓ→₀⁻ f(x) = 1

and

limₓ→₀⁺ f(x) = 1.

Here both 0⁺ and 0⁻  are called right and left limits. Both the right and left limits approaches 0 the value of f(x) = 1.

When it is different from different sides:

When the function f(x) breaks the limit does not exists at that point.

We can say from the following diagram, that the function exists at x = a.

Here the limit of the function does not exist at x = a where a=5.

Here the left hand side is 3.8

Right hand side is 1.2.

Limit  approaching infinity:

Now let us see the value of function as x approaches ∞.

Let us see the function f(x) = 1/x.  Now let us see the value of f(x)  as x approaches ∞.

Now as x becomes larger,  the value of 1/x tends to 0.