This page contains peer generated solutions and error explanations to OFSA questions. As you read or view the solutions, be critical: check for accuracy, but also for more efficient solution strategies. If you have a better method or different idea/answer, post a discussion and monitor the responses.

Quick Directions

Post answers, solutions and error explanations to each OFSA question below.

For each "distractor" or incorrect answer choice, explain the error that would lead to that incorrect answer choice.

You may either do the above in typed format or using a pencast.

Separate each question with a section bar.

After each solution, provide a hyperlink back to the corresponding OFSA page.

A. 6x+3h-5 Error. Student forgot to evaluate the limit as h approaches the value 0 although they may have simplified the function to its simplest form (they forgot to find the limit using direct substitution.)

B. 6x-5 Correct!!!

C. 0 Error. Student only focused on manipulating the algebra in the numerator that they forgot to divide the entire numerator by h.

D. Does Not Exist (DNE) Error. Student forgot to distribute the negative when subtracting the entire f(x) function from the f(x+h) function.

B. 1 Error. Student simply used direct substitution to solve the problem and got

Then, the student assumed that the result of this expression is 1. However, they do not realize that different levels of infinity exist.

C. 2/5 Error. Student thought

when really

for limits at infinity problems.

D. Indeterminant Error. Student simply used direct substitution to solve the problem and got

like in B. However, this student assumed that the result of this expression was indeterminant form and couldn't be taken any further.(So there could be two different interpretations of this expression.)

A. f(a) does not exist, so therefore the function is not continuous
B. 5
C. Undefined
D. 0

Solution 3

Error Explanation 3

A. f(a) does not exist, so therefore the function is not continuous Error. Although the student demonstrates their understanding of the definition of continuity [f(a) must exist], they don't realize that f(-2) does, in fact, exist because the restrictions accommodate for the value of -2 from the left, at -2, and from the right.

B. 5 Error. Student only evaluated the first equation by substituting -2 in for x and thought this was the value of A that would make the function continuous.

C. Undefined Error. Student plugged in the value of -2 for Ainstead of x, in which case they got an expression "1=5" which is not true and would lead them to believe the answer is undefined.

Solution 4 Error Explanation 4
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Question 5

Solution 5 Error Explanation 5
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Question 6

Solution 6 Error Explanation 6
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Question 7

Solution 7 Error Explanation 7
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Question 8

Solution 8 Error Explanation 8
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Question 9

Solution 9 Error Explanation 9
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Question 10

Solution 10 Error Explanation 10
Click here to return to OFSA. Question 11 Solution 11 Error Explanation 11
Click here to return to OFSA. Question 12 Solution 12 Error Explanation 12
Click here to return to OFSA. Question 13 Solution 13

a. 1
b. -3/2
c. DNE
d. 2/3

Step One:

Step 2

Remember,

is very to close 0, so we can count it as zero. IT CANNOT ACTUALLY BE ZERO

Error Explanation 13

a. 1 This is the correct answer :)
b. -3/2 A student would get this answer if they were to substitute zero for x, forgetting to divide by the highest bottom exponent of X
c. DNE A student would get this if they incorrectly assume that one cannot have a limit at infinity.
d. 2/3 A student would get this if they substituted for 1 for x

Error Explanation 14
a.13- A student substitutes 0 in for X instead of 4.
b. 15/16- If a student were rushing and were to add the 3 instead of subtract it, they would get this answer.
c. 9/16 - correct answer
d. Never Continuous- If a student incorrectly assumes that two equations will never meet, then it should be never continuous. This will never happen unless the all the lines given are parallel.

c.-1/9
d. DNE Solution 15
DOES NOT EXIST. Because it cannot be canceled out anywhere, it is asymptote that has 2 different Y-values as X approaches 3 from the positive and negative. Error Explanation 15
a. 1/9- Often, if a student has a DNE answer, they doubt themselves and will go looking for the best other option. This one is the opposite of answer C, so then they would look at this answer and begin to doubt what they know is true. This is a very blatant distractor.
b.

- If the student were to be solving

then they would get three as an answer.
c.-1/9- A student would get this answer if they were to either substitute zero in, or if they were to disregard the X in total.
d. DNE- Correct answer, see explanation above

Error Explanation 16
a. DNE- Student assumed that the limit doesn't exist because he only paid attention to the x in the denominator
b. 1/3- Student was on the right track but, at the end, made a simple mistake by substituting the whole expression in the square instead of just x squared
c. 1/6- Correct answer
d. 1/12- Student was on the right track but, at the end, made a simple mistake by forgetting about the square root
e. none of the above- Student did something else leading to an incorrect answer Click here to return to OFSA. Question 17

a. DNE
b.

c.

d.

e. none of the above

Solution 17

Error Explanation 17
a. DNE- Student assumed that the limit doesn't exist because he only paid attention to the x in the denominator
b.

Student was close but saw only one square root of 2 when there were 2 square root of 2's
c.

same as b because they are the same answer
d.

correct answer
e. none of the above- Student did something else leading to an incorrect answer

0 is neither positive nor negative, so in this case, absolute value doesn't matter

Error Explanation 18
a. DNE- Student probably guessed
b. 1- Student probably guessed
c. 0- correct answer
d. -1- Student probably guessed
e. none of the above- Student did something else leading to an incorrect answer

Finding the left and right side limits is the best thing to do in this problem

Error Explanation 19
a. DNE- correct answer
b. 1- Student had the right idea but had mistaken the right side limit as the answer
c. 0- Student either forgot about x in denominator or guessed
d. -1- Student had the right idea but had mistaken the left side limit as the answer
e. none of the above- Student did something else leading to an incorrect answer

Click here to return to OFSA.
Question 20 Solution 20 Error Explanation 20 Click here to return to OFSA.
Question 21 Solution 21 Error Explanation 21 Click here to return to OFSA.
Question 22 Solution 22 Error Explanation 22 Click here to return to OFSA.
Question 23 Solution 23 Error Explanation 23 Click here to return to OFSA.
Question 24 Solution 24 Error Explanation 24**
Click here to return to OFSA. Student did something else leading to an incorrect answer

OFSA SOLUTIONS for LimitsThis page contains peer generated solutions and error explanations to OFSA questions. As you read or view the solutions, be critical: check for accuracy, but also for more efficient solution strategies. If you have a better method or different idea/answer, post a discussion and monitor the responses.

Quick DirectionsQuestion 1## Table of Contents

evaluate

A. 6x+3h-5

B. 6x-5

C. 0

D. Does Not Exist (DNE)

Solution 1Error Explanation 1A. 6x+3h-5

Error. Student forgot to evaluate the limit as h approaches the value 0 although they may have simplified the function to its simplest form (they forgot to find the limit using direct substitution.)B. 6x-5

Correct!!!C. 0

Error. Student only focused on manipulating the algebra in the numerator that they forgot to divide the entire numerator by h.D. Does Not Exist (DNE)

Error. Student forgot to distribute the negative when subtracting the entire f(x) function from the f(x+h) function.Click here to return to OFSA.

Question 2Evaluate

A. 1/4

B. 1

C. 2/5

D. Indeterminant

Solution 2Error Explanation 2A. 1/4

Correct!!!B. 1

Error. Student simply used direct substitution to solve the problem and gotThen, the student assumed that the result of this expression is 1. However, they do not realize that different levels of infinity exist.C. 2/5

Error. Student thoughtwhen reallyfor limits at infinity problems.D. Indeterminant

Error. Student simply used direct substitution to solve the problem and gotlike inB.However, this student assumed that the result of this expression was indeterminant form and couldn't be taken any further.(So there could be two different interpretations of this expression.)Click here to return to OFSA.

Question 3Find the value of A that makes

continuous.

A. f(a) does not exist, so therefore the function is not continuous

B. 5

C. Undefined

D. 0

Solution 3Error Explanation 3A. f(a) does not exist, so therefore the function is not continuous

Error. Although the student demonstrates their understanding of the definition of continuity [f(a) must exist], they don't realize that f(-2) does, in fact, exist because the restrictions accommodate for the value of -2 from the left, at -2, and from the right.B. 5

Error. Student only evaluated the first equation by substituting -2 in for x and thought this was the value of A that would make the function continuous.C. Undefined

Error. Student plugged in the value of -2 for Ainstead of x, in which case they got an expression"1=5"which is not true and would lead them to believe the answer is undefined.D. 0

Correct!!!Click here to return to OFSA.

Question 4Solution 4Error Explanation 4Click here to return to OFSA.

Question 5Solution 5Error Explanation 5Click here to return to OFSA.

Question 6Solution 6Error Explanation 6Click here to return to OFSA.

Question 7Solution 7Error Explanation 7Click here to return to OFSA.

Question 8Solution 8Error Explanation 8Click here to return to OFSA.

Question 9Solution 9Error Explanation 9Click here to return to OFSA.

Question 10Solution 10Error Explanation 10Click here to return to OFSA.

Question 11Solution 11Error Explanation 11Click here to return to OFSA.

Question 12Solution 12Error Explanation 12Click here to return to OFSA.

Question 13Solution 13a. 1b. -3/2

c. DNE

d. 2/3

Step One:

Step 2

Remember,

is very to close 0, so we can count it as zero. IT CANNOT ACTUALLY BE ZERO

Error Explanation 13a. 1 This is the correct answer :)b. -3/2 A student would get this answer if they were to substitute zero for x, forgetting to divide by the highest bottom exponent of X

c. DNE A student would get this if they incorrectly assume that one cannot have a limit at infinity.

d. 2/3 A student would get this if they substituted for 1 for x

Click here to return to OFSA.Question 14

a.13

b.15/16

c. 9/16

d.Never can be continuous

Solution 14Step 1:

Step 2:

Step 3:

Step 4:

Error Explanation 14

a.13- A student substitutes 0 in for X instead of 4.

b. 15/16- If a student were rushing and were to add the 3 instead of subtract it, they would get this answer.

c. 9/16 - correct answer

d. Never Continuous- If a student incorrectly assumes that two equations will never meet, then it should be never continuous. This will never happen unless the all the lines given are parallel.

Click here to return to OFSA.Question 15

a.1/9

b.

c.-1/9

d. DNE

Solution 15DOES NOT EXIST. Because it cannot be canceled out anywhere, it is asymptote that has 2 different Y-values as X approaches 3 from the positive and negative.

Error Explanation 15a. 1/9- Often, if a student has a DNE answer, they doubt themselves and will go looking for the best other option. This one is the opposite of answer C, so then they would look at this answer and begin to doubt what they know is true. This is a very blatant distractor.

b.

- If the student were to be solving

then they would get three as an answer.

c.-1/9- A student would get this answer if they were to either substitute zero in, or if they were to disregard the X in total.

d. DNE- Correct answer, see explanation above

Click here to return to OFSA.Question 16a. DNE

b. 1/3

c. 1/6

d. 1/12

e. none of the above

Solution 16Error Explanation 16a. DNE- Student assumed that the limit doesn't exist because he only paid attention to the x in the denominator

b. 1/3- Student was on the right track but, at the end, made a simple mistake by substituting the whole expression in the square instead of just x squared

c. 1/6- Correct answer

d. 1/12- Student was on the right track but, at the end, made a simple mistake by forgetting about the square root

e. none of the above- Student did something else leading to an incorrect answer

Click here to return to OFSA.Question 17a. DNE

b.

c.

d.

e. none of the above

Solution 17Error Explanation 17a. DNE- Student assumed that the limit doesn't exist because he only paid attention to the x in the denominator

b.

Student was close but saw only one square root of 2 when there were 2 square root of 2's

c.

same as b because they are the same answer

d.

correct answer

e. none of the above- Student did something else leading to an incorrect answer

Click here to return to OFSA.Question 18a. DNE

b. 1

c. 0

d. -1

e. none of the above

Solution 180 is neither positive nor negative, so in this case, absolute value doesn't matter

Error Explanation 18a. DNE- Student probably guessed

b. 1- Student probably guessed

c. 0- correct answer

d. -1- Student probably guessed

e. none of the above- Student did something else leading to an incorrect answer

Click here to return to OFSA.Question 19a. DNE

b. 1

c. 0

d. -1

e. none of the above

Solution 19Finding the left and right side limits is the best thing to do in this problem

Error Explanation 19a. DNE- correct answer

b. 1- Student had the right idea but had mistaken the right side limit as the answer

c. 0- Student either forgot about x in denominator or guessed

d. -1- Student had the right idea but had mistaken the left side limit as the answer

e. none of the above- Student did something else leading to an incorrect answer

Click here to return to OFSA.Question 20

Solution 20Error Explanation 20Click here to return to OFSA.Question 21

Solution 21Error Explanation 21Click here to return to OFSA.Question 22

Solution 22Error Explanation 22Click here to return to OFSA.Question 23

Solution 23Error Explanation 23Click here to return to OFSA.Question 24

Solution 24Error Explanation 24**Click here to return to OFSA.

Student did something else leading to an incorrect answer