Question

Find all x-values where f(x) is discontinuous. For each discontinuity, state which of the three conditions is not fulfilled.

1) f(x) ={ (2x^2 -x-3)/5  if x does not equal to -1

                5                        if x = -1

 

2) f(x) = { (x-2) / (x^2 -4)      if x < -3

                 5-x-x^2                  if -3 less than or equal to x < 0

                  5sin^2(x+(pi/2))   if x > 0

Answer 1

Step 1:

(1)

The function is

 

Definition of Continuity:

A function is continuous at a number , if

(1) is defined.

(2) exists.

(3) .

Consider

and .

.

The function is discontinuous at .

Solution:

The function is discontinuous at .

Answer 2

Step 1:

(2)

The function is

 

Definition of Continuity:

A function is continuous at a number ,

(1) is defined.

(2) exists.

(3) .

 The function is continuous for all values of .

 The function is continuous for all values of .

The function is continuous for all values of .

Step 2:

Consider left hand limit at .

Cancel the common terms.

Consider right hand limit at .

As the right hand limit and the left hand limit are equal, limit exists at .

.

Comments

Step 3:

Consider left hand limit at .

Consider right hand limit at .

As the right hand limit and the left hand limit are equal, limit exists at .

.

The function is continuous for all values of .

Solution:

The function is continuous for all values of .