<< Chapter < Page Chapter >> Page >

f ( x ) = { 5 ,    x 0 3 ,    x = 0    a = 0

Got questions? Get instant answers now!

f ( x ) = { 1 2 x , x 2 3 , x = 2    a = 2

lim x 2 f ( x ) does not exist.

Got questions? Get instant answers now!

f ( x ) = { 1 x + 6 , x = 6 x 2 , x 6    a = 6

Got questions? Get instant answers now!

f ( x ) = { 3 + x , x < 1 x , x = 1 x 2 , x > 1      a = 1

lim x 1 f ( x ) = 4 ; lim x 1 + f ( x ) = 1 . Therefore, lim x 1 f ( x ) does not exist.

Got questions? Get instant answers now!

f ( x ) = { 3 x , x < 1 x , x = 1 2 x 2 , x > 1      a = 1

Got questions? Get instant answers now!

f ( x ) = { 3 + 2 x , x < 1 x , x = 1 x 2 , x > 1      a = 1

lim x 1 f ( x ) = 5 lim x 1 + f ( x ) = 1 . Thus lim x 1 f ( x ) does not exist.

Got questions? Get instant answers now!

f ( x ) = { x 2 , x < 2 2 x + 1 , x = 2 x 3 , x > 2      a = 2

Got questions? Get instant answers now!

f ( x ) = { x 2 9 x + 3 , x < 3 x 9 , x = 3 1 x , x > 3      a = 3

lim x 3 f ( x ) = 6 , lim x 3 + f ( x ) = 1 3

Therefore, lim x 3 f ( x ) does not exist.

Got questions? Get instant answers now!

f ( x ) = { x 2 9 x + 3 , x < 3 x 9 , x = 3 6 , x > 3      a = 3

Got questions? Get instant answers now!

f ( x ) = x 2 4 x 2 ,    a = 2

f ( 2 ) is not defined.

Got questions? Get instant answers now!

f ( x ) = 25 x 2 x 2 10 x + 25 ,    a = 5

Got questions? Get instant answers now!

f ( x ) = x 3 9 x x 2 + 11 x + 24 ,    a = 3

f ( 3 ) is not defined.

Got questions? Get instant answers now!

f ( x ) = x 3 27 x 2 3 x ,    a = 3

Got questions? Get instant answers now!

f ( x ) = x | x | ,    a = 0

f ( 0 ) is not defined.

Got questions? Get instant answers now!

f ( x ) = 2 | x + 2 | x + 2 ,    a = 2

Got questions? Get instant answers now!

For the following exercises, determine whether or not the given function f is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.

f ( x ) = x 3 2 x 15

Continuous on ( , )

Got questions? Get instant answers now!

f ( x ) = x 2 2 x 15 x 5

Got questions? Get instant answers now!

f ( x ) = 2 3 x + 4

Continuous on ( , )

Got questions? Get instant answers now!

f ( x ) = −sin ( 3 x )

Got questions? Get instant answers now!

f ( x ) = | x 2 | x 2 2 x

Discontinuous at x = 0 and x = 2

Got questions? Get instant answers now!

f ( x ) = 2 x + 5 x

Discontinuous at x = 0

Got questions? Get instant answers now!

f ( x ) = ln   x 2

Continuous on ( 0 , )

Got questions? Get instant answers now!

f ( x ) = x 4

Continuous on [ 4 , )

Got questions? Get instant answers now!

f ( x ) = sec ( x ) 3 .

Got questions? Get instant answers now!

f ( x ) = x 2 + sin ( x )

Continuous on ( , ) .

Got questions? Get instant answers now!

Determine the values of b and c such that the following function is continuous on the entire real number line.

f ( x ) = { x + 1 , 1 < x < 3 x 2 + b x + c , | x 2 | 1 }

Got questions? Get instant answers now!

Graphical

For the following exercises, refer to [link] . Each square represents one square unit. For each value of a , determine which of the three conditions of continuity are satisfied at x = a and which are not.

Graph of a piecewise function where at x = -3 the line is disconnected, at x = 2 there is a removable discontinuity, and at x = 4 there is a removable discontinuity and f(4) exists.

x = 3

1, but not 2 or 3

Got questions? Get instant answers now!

x = 4

1 and 2, but not 3

Got questions? Get instant answers now!

For the following exercises, use a graphing utility to graph the function f ( x ) = sin ( 12 π x ) as in [link] . Set the x -axis a short distance before and after 0 to illustrate the point of discontinuity.

Graph of the sinusodial function with a viewing window of [-10, 10] by [-1, 1].

Which conditions for continuity fail at the point of discontinuity?

Got questions? Get instant answers now!

Evaluate f ( 0 ) .

f ( 0 ) is undefined.

Got questions? Get instant answers now!

Solve for x if f ( x ) = 0.

Got questions? Get instant answers now!

What is the domain of f ( x ) ?

( , 0 ) ( 0 , )

Got questions? Get instant answers now!

For the following exercises, consider the function shown in [link] .

Graph of a piecewise function where at x = -1 the line is disconnected and at x = 1 there is a removable discontinuity.

At what x -coordinates is the function discontinuous?

Got questions? Get instant answers now!

What condition of continuity is violated at these points?

At x = 1 , the limit does not exist. At x = 1 , f ( 1 ) does not exist.

At x = 2 , there appears to be a vertical asymptote, and the limit does not exist.

Got questions? Get instant answers now!

Consider the function shown in [link] . At what x -coordinates is the function discontinuous? What condition(s) of continuity were violated?

Graph of a piecewise function where at x = -1 the line is disconnected and where at x = 1 and x = 2 there are a removable discontinuities.
Got questions? Get instant answers now!

Construct a function that passes through the origin with a constant slope of 1, with removable discontinuities at x = 7 and x = 1.

x 3 + 6 x 2 7 x ( x + 7 ) ( x 1 )

Got questions? Get instant answers now!

The function f ( x ) = x 3 1 x 1 is graphed in [link] . It appears to be continuous on the interval [ 3 , 3 ] , but there is an x -value on that interval at which the function is discontinuous. Determine the value of x at which the function is discontinuous, and explain the pitfall of utilizing technology when considering continuity of a function by examining its graph.

Graph of the function f(x) = (x^3 - 1)/(x-1).
Got questions? Get instant answers now!

Find the limit lim x 1 f ( x ) and determine if the following function is continuous at x = 1 :

f x = { x 2 + 4 x 1 2 x = 1

The function is discontinuous at x = 1 because the limit as x approaches 1 is 5 and f ( 1 ) = 2.

Got questions? Get instant answers now!

The graph of f ( x ) = sin ( 2 x ) x is shown in [link] . Is the function f ( x ) continuous at x = 0 ? Why or why not?

Graph of the function f(x) = sin(2x)/x with a viewing window of [-4.5, 4.5] by [-1, 2.5]
Got questions? Get instant answers now!
Practice Key Terms 4

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask