help?CONTINUITY

Question

Answer 1

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

(1)  is defined.

(2)  exists.

(3) .

Observe the graph:

  is discontinuous at , and .

Check the continuity at .

The value of the function at is .

is defined at .

Comments

Contd..

Left hand limit :

As tends to -2 from left hand side, approaches to 3.

.

Right hand limit :

As tends to -2 from right hand side, approaches to 3.

.

As left hand limit and the right hand limits are equal, limit exist.

is exist.

Hence , .

fails the continuity test condition (3) at .

is discontinuous at .

Solution:

is discontinuous at .

Answer 2

Check for the continuity at .

Observe the graph:

The value of the function at is .

is defined at .

Left hand limit :

As tends to 1 from left hand side, approches to -3.

.

Right hand limit :

As tends to 1 from right hand side, approches to 4.
.

Since , does not exist.

fails the continuity test conditions (2), (3) at .

 is discontinuous at .

Solution:

 is discontinuous at .

Answer 3

Check for the continuity at .

The value of the function at is undefined.

is undefined at .

Left hand limit :

As tends to 4 from left hand side, approaches to .

.

Right hand limit :

As tends to 4 from right hand side, approaches to .

.

Since , does not exist.

fails the continuity test condition (1), (2), (3) at .

is discontinuous at .

Solution:

  is discontinuous at .

  is discontinuous at , and .