OFSA SOLUTIONS for Limits

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.
  • Follow example below.
  • Click here to refer to solution format in 7.7



Question 1

Given

evaluate


A. 6x+3h-5
B. 6x-5
C. 0
D. Does Not Exist (DNE)

Solution 1

















Error Explanation 1

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.

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Question 2


Evaluate


A. 1/4
B. 1
C. 2/5
D. Indeterminant

Solution 2







Error Explanation 2

A. 1/4 Correct!!!

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.)

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Question 3


Find 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 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 A instead 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!!!

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Question 4

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
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Question 11
Solution 11
Error Explanation 11
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Question 12
Solution 12
Error Explanation 12
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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



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Question 14


a.13
b.15/16
c. 9/16
d.Never can be continuous

Solution 14
Step 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.

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Question 15

a.1/9
b.

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

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Question 16



a. DNE
b. 1/3
c. 1/6
d. 1/12
e. none of the above

Solution 16









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
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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

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Question 18



a. DNE
b. 1
c. 0
d. -1
e. none of the above

Solution 18


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

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Question 19



a. DNE
b. 1
c. 0
d. -1
e. none of the above

Solution 19

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

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Question 20
Solution 20
Error Explanation 20
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Question 21
Solution 21
Error Explanation 21
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Question 22
Solution 22
Error Explanation 22
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Question 23
Solution 23
Error Explanation 23
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Question 24
Solution 24
Error Explanation 24**
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Student did something else leading to an incorrect answer