The limit of a function is the most fundamental concept in the study of Calculus as well as of mathematical analysis.

Limit of a function at a point is defined by where is read as approaches (or tends) to and .

Consider a function defined by .

The domain of this function is the set of real numbers. .

It is natural to ask whether or not becomes closer and closer to as becomes closer and closer to but not equal to .

Hence i.e., as

The statement is also expressed by saying that limit of at is equal to

__Algebra of Limits:__

Let and be two functions such that both and exists , then

- If be a constant function such that , for some real number .

__Example 1:__

Find

__Solution:__

__Example 2:__

Find

__Solution:__

__Example 3:__

A point is at a distance less than from the point , then the set of all such that is at a distance less than any fixed from is expressed as

“ is arbitrarily close to and ”.

__The neighborhood of a Point :__

If , then any open interval containing the point is called a neighborhood of a point.

The open interval is called the – neighborhood of the point where .

(iii)

(iv)

__Definition of Limit of Function:__

A number is called the limit of a function as tends to i.e. if for any there exist depending on such that :

i.e. and

__Example:__

__Solution:__

Let , we take then if taking we see that there exists depending in such that:

So,

Hence

i.e.,

__Example 1:__

Solve:

__Solution:__

__Example 2:__

__Example 3:__

__Left hand and Right-hand Limit of a Function__

For any there exists depending on such that and , then

and means or

So if (i) and (ii)

then

__Left hand limit__

A number is called the left-hand limit of at or simply if for any there exists depending on such that

__Right-hand limit:__

A number is called the right-hand limit of at it for any there exists depending in such that i.e.

__Note-:__

(i) If

i.e. , then exists and equal to

i.e.

(ii) If , then does not exist.

(iii) For a function to have a limit at a point it is necessary and sufficient that both and exist, and these limits coincide.

__Infinite limits__

(i) ,if given , there exists depending on such that .

(ii) , if given there exists depending on such that where is as large as possible number greater than .

__Limits at Infinity:__

(i) if given , there exists depending on such that

(ii) if given , there exists depending on such that

(iii) if given , there exists such that .The concepts and are also defined by infinite limits at infinity.

__Example:__

Find

__Solution:__

We choose , then if we take .

Thus given , there exists depending on such that .

So as

i.e.

__Note:__

If increases indefinitely through positive values, remains positive and decreases indefinitely. So intuitively we see that as .

__Note:__

and can be written as and in that case , holds if is replaced by and in second case is replaced by

__Theorem:__

(i) If a function satisfies the inequality in a neighborhood of and a function is such that , for and .

(ii) If in a neighborhood of the point and if is such that , for and then

__Sandwich theorem / Squeezing theorem:__

If and a function is such that for all in a deleted neighborhood of , then .

__Example:__

If , then find by sandwich theorem.

__Solution:__

Since , with , we have and so by Sandwich Theorem:

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