help-Continuity

Question

Answer 1

(2_a)

Step 1:

The function is and .

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.

The function is .

Substitute in the function.

.

is defined.

Step 2:

Condition (2): exists.

.

exists.

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 2

(2_b)

Step 1:

The function is and .

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.

The function is .

Substitute in the function.

.

is defined.

Step 2:

Condition (2): exists.

.

exists.

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

(2_c)

Step 1:

The function is and .

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.

The function is .

Substitute in the function.

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 .

Answer 4

(3_a)

Step 1:

The function is and .

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.

If then .

Substitute in above function.

.

is defined.

Step 2:

Condition (2): exists.

.

exists.

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 5

(3_b)

Step 1:

The function is and .

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.

If then .

Substitute in above function.

.

is defined.

Step 2:

Condition (2): exists.

Left hand limit :

If then .

.

Right hand limit :

If then .

.

Comments

Contd....

Left hand limit and right hand limit are not equal, limit does not exist.

does not exists.

Here limit does not exist at .

The condition (2) of continuity is not satisfied, hence the function is discontinuous.

The function is discontinuous at .

Solution :

The function is discontinuous at .

Answer 6

(4_a)

Step 1:

The function is and .

Simplify the function.

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.

The function is .

Substitute in the function.

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 .

Answer 7

(4_b)

Step 1:

The function is and .

Simplify the function.

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.

The function is .

Substitute in the function.

.

 is defined.

Step 2:

Condition (2): exists.

.

exists.

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 8

(4_c)

Step 1:

The function is and .

Simplify the function.

Definition of continuity :

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

(1) is defined.

(2) exists.

(3) .

Step 2:

Condition (1): is defined.

The function is .

Substitute in the function.

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 .