Continuity Problems

Solutions

1Find the point(s) of discontinuity for the function f(x) = x2 + 1+ |2x − 1|.

2Consider the function:

Continuity Exercise

If f (2) = 3, determine the values of a and b for which f(x) is continuous.

3Given the function:

Continuity Exercise

Determine the value of a for which the function is continuous at x = 3.

4Given the function:

Continuity Exercise

Determine the points of discontinuity.

5Given the function:

Continuity Exercise

Determine a and b so that the function f(x) is continuous for all values of x.

6Given the function:

Continuity Exercise

Determining the value of a for which f(x) is continuous.

7Calculate the value of k for the following continuous function.

Continuity Exercise

8Given the function:

Determine the values for a and b in order to create a continuous function.

9Determine the values for a and b in order to create a continuous function.

Continuity Exercise


1

Find the point(s) of discontinuity for the function f(x) = x2 + 1+ |2x − 1|.

Continuity Problem

Continuity Operations

Continuity Operations

Continuity Operations

Continuity Operations

Continuity Solution

There are no points of discontinuity as the function is continuous.


2

Consider the function:

Continuity Problem

If f (2) = 3, determine the values of a and b for which f (x) is continuous.

There is only a question of continuity at x = 1.

Continuity Operations

Continuity Operations

Continuity Operations

For the function to be continuous:

Continuity Operations

On the other hand there is:

Continuity Solution

Solve the system of equations and obtain:

a = 1 b = −1


3

Given the function:

Continuity Problem

Determine the value of a for which the function is continuous at x = 3.

Continuity Operations

Continuity Operations

Continuity Solution


4

Given the function:

Continuity Problem

Determine the points of discontinuity for the function.

The exponential function is positive for all x pertenece R, therefore the denominator of the function cannot be annulled.

There is only doubt of the continuity at x = 0.

Continuity Operations

Continuity Operations

Solve the indeterminate form dividing by Continuity Operations

Continuity Operations

The function is continuous on R − {0}.

See also in trigonometric working demo.


5

Given the function:

Continuity Problem

Determine a and b so that the function f(x) is continuous for all values of x.

Continuity Operations

Continuity Operations

Continuity Operations

Continuity Operations

Continuity Operations

Continuity Operations

Continuity Solution


6

Given the function:

Continuity Problem

Determining the value of a for which f(x) is continuous.

Continuity Operations

Continuity Operations

Continuity Solution


7

Calculate the value of k for the following continuous function.

Continuity Problem

Continuity Operations

Continuity Solution

Therefore there is no limit for the function and there is no value that would make f(x) continuous at x = 0, regardless of what value k is given.


8

Given the function:

Determine the values for a and b in order to create a continuous function.

Continuity Operations

Continuity Operations

Continuity Operations

Continuity Operations

Continuity Operations

Continuity Operations

Continuity Operations

Continuity Solution


9

Determine the values for a and b in order to create a continuous function.

Continuity Problem

b= 1

Continuity Operations

Continuity Solution

3a = −2 a = −1