### Mathematics Class XI

Unit-I: Sets and Functions
Chapter 1: Sets
Unit-II: Algebra
Chapter 5: Binomial Theorem
Chapter 6: Sequence and Series
Unit-III: Coordinate Geometry
Chapter 1: Straight Lines
Chapter 2: Conic Sections
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability
Chapter 1: Statistics
Chapter 2: Probability

# Limit of functions

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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