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Question

Answer 1

(11)

Step 1 :

Definition of continuity :

A function is continuous at , if then it should satisfy three conditions :

(1) is defined.

(2) exists.

(3) .

Condition (1): is defined.

Observe the Graph :

At point .

.

is defined.

Step 2:

Condition (2): exists.

Observe the graph.

Left hand limit :

As x tends to from left side, approaches to .
.

Right hand limit :

As x tends to from right side, approaches to .

.

Left hand limit and right hand limit are equal, limit exist.

exist.

Comments

contd....

Step 3:

Condition (3): .

and .

.

The condition (3) of continuity is not satisfied, hence the function is discontinuous.

The function is discontinuous at .

Solution :

The function is discontinuous at .

Answer 2

 (12)

Step 1 :

Definition of continuity :

A function is continuous at , if then it should satisfy three conditions :

(1) is defined.

(2) exists.

(3) .

Condition (1): is defined.

Observe the Graph :

At , .

.

is defined.

Step 2 :

Condition (2): exists.

Observe the graph.

Left hand limit :

As x tends to from left side, approaches to .
.

Right hand limit :

As x tends to from right side, approaches to .

.

Left hand limit and right hand limit are equal, limit exist.

.

exist.

Comments

Contd......

Step 3 :

Condition (3): .

and .

.

The three conditions of continuity are satisfied, hence the function is continuous.

The function is continuous at .

Solution :

The function is continuous at .

Answer 3

(13)

Step 1 :

Definition of continuity :

A function is continuous at , if then it should satisfy three conditions :

(1) is defined.

(2) exists.

(3) .

Condition (1): is defined.

Observe the Graph :

At , is undefined.

is undefined.

The condition (1) of continuity is not satisfied, hence the function is discontinuous.

The function is discontinuous at .

Solution :

The function is discontinuous at .