FINDING THE POINT OF DISCONTINUITY WORKSHEET

Find any points of discontinuity for each rational function.

Problem 1 :

y = (x + 3)/(x – 4) (x + 3)

Solution

Problem 2 :

y = (x - 2)/(x² – 4)

Solution

Problem 3 :

y = (x - 3) (x + 1)/(x – 2)

Solution

Problem 4 :

y = 3x(x + 2)/x(x + 2)

Solution

Problem 5 :

y = 2/(x + 1)

Solution

Problem 6 :

y = 4x/(x³ – 9x)

Solution

Problem 7 :

Check the function

g(x) = (x² + 7x + 10)/(x - 3)(x + 2)

for discontinuities. Conduct appropriate tests to determine if asymptotes exist at the discontinuity values. State the equations of any asymptotes and the domain of g(x)

Solution

Problem 8 :

Find the points of discontinuity of the function and identify the type of discontinuity.

f(x) = (x² - 4x + 3)/(x - 1)

Solution

Problem 9 :

Find the value of k in each of the following:

is continuous at x = 5

Solution

Problem 10 :

Use the function f defined and graphed below to answer the questions.

a) Does f(-1) exists ?

b) Does lim x-> 1^+ f(x) exists ?

c)  Does lim x-> 1^+ f(x) = f(-1)

d) Is f continuous at x = -1

Solution

Problem 1 :

Match each function with its graph.

a) h(x) = (x + 4) / (2x + 5)

b) f(x) = 3/(x - 1)

c) m(x) = (2x -  4) / (x - 2)

d) g(x)= (2x - 3) / (x + 2)

Solution

Problem 2 :

Write an equation for a rational function with the properties as given.

a) a hole at x = 1

b) a vertical asymptote anywhere and a horizontal asymptote along the x-axis

c) a hole at x = -2 and a vertical asymptote at x = 1 

d) a vertical asymptote at x = 1and a horizontal asymptote at y = 2

e) an oblique asymptote, but no vertical asymptote

Solution

Discuss the continuity. If a discontinuity exists, then describe the type of discontinuity and its physical feature on a graph.

Problem 1 :

Solution

Problem 2 :

Solution

Problem 3 :

Solution

Problem 4 :

Solution

Problem 5 :

Solution

Problem 6 :

Solution

Problem 7 :

For which of the following does lim x--> 4 exists ?

a) I only     b) II only     c) III only     d) I and II only      e)  I and III only

Solution

Problem 8 :

For the function identify the type of each discontinuity and where it is located.

f(x) = (x² - 8x + 12)/(x² + 3x - 10)

Solution

Problem 9 :

Determine if each function is continuous. If the function is not continuous, find the x-axis location of and classify each discontinuity.

f(x) = -x³ + x² - 3

Solution

Problem 10 :

f(x) = -(x + 2) / (x² - 4)

Solution

Problem 11 :

Solution

Problem 12 :

The graph of the function 𝑓(𝑥) is shown to the right: Which of the following statements is true about 𝑓?

I. 𝑓 is undefined at 𝑥 = 1.

II. 𝑓 is defined but not continuous at 𝑥 = 2.

III. 𝑓 is defined and continuous at 𝑥 = 3.

(A) Only I     (B) Only II     (C) I and II     (D) I and III

(E) None of the statements are true.

Solution