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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 ₓ₋ₐ 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.

We had seen introduction and definition in this page. More problems we will discuss in the other pages. You can go through the problems and if you have any doubt you can contact us through mail, we will help you to clear your doubts.

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