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NJCTL.org
LIMITS UNIT PROBLEM SETS
PROBLEM SET #1 β Tangent Lines ***Calculators Not Allowed***
1. Given: π(π₯) = 2π₯ β 7 Find the slope of the tangent line at π₯0 = 3
2. Given: π(π₯) = β4π₯ β 2 Find the slope of the secant line between π₯1 = β2
and π₯2 = 3
3. Given: π(π₯) = 4π₯2 + 7 Find the equation of the tangent line at π₯0 =1
2
4. Given: π(π₯) = β2π₯2 β 3π₯ Find the slope of the tangent line at π₯0 = β2
5. Given: π(π₯) = π₯3 + 8 Find the slope of the secant line between π₯1 = 0 and π₯2 = 1
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6. Given: π(π₯) = 2π₯3 + π₯ Find the equation of the tangent line at π₯0 = 0
7. Given: π(π₯) = 7 Find the slope of the tangent line at π₯0 = 1
8. Given: π(π₯) =2
3π₯ + 4 Find the slope of the secant line between π₯1 = 6 and
π₯2 = 9
9. Given: π(π₯) = 2π₯2 β 10 Find the equation of the tangent line at π₯0 = 3
10. Given: π(π₯) = β1
π₯ Find the slope of the tangent line at π₯0 = 2
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PROBLEM SET #2 β Limits (Graphs) ***Calculators Not Allowed***
For problems #1-8, find the limit of the function at the given point:
1. lim
π₯β0π(π₯) = ________________
2. limπ₯β2
π(π₯) = ________________
3. limπ₯ββ1β
π(π₯) = ________________
4. limπ₯ββ1+
π(π₯) = ________________
Use for problems #1-4
5. limπ₯ββ1β
π(π₯) = ________________
6. limπ₯ββ1+
π(π₯) = ________________
7. limπ₯β3β
π(π₯) = ________________
8. limπ₯β3+
π(π₯) = ________________
Use for problems #5-8
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PROBLEM SET #3 β Computing Limits ***Calculators Not Allowed***
For the following, find the limit of the function at the given point:
1. limπ₯β
π2
3sin π₯
2. limπ₯β
12
(β4π₯ + 2)
3. limπ₯β3
(β3π₯2 + 7π₯)
4. lim π₯β0
π₯(3π₯2 + 7)
5. limπ₯βπ
ln|π₯|
6. limπ₯βπ
ln|3π₯|
7. limπ₯β3
|π₯4 β 2π₯3 β 30|
8. limπ₯β3+
β(π₯2 β 9)
9. limπ₯β4β
β(π₯2 β 16)
10. limπ₯β3
(π₯ + 2)(π₯ β 3)
11. limπ₯ββ2
(β3π₯3 + 4π₯2 β 10)
12. limπ₯β
π2
cot π₯
13. limπ₯ββ5
β(π₯ + 4)
14. limπ₯βπ
2 cos(2π₯)
15. limπ₯β0
ln|2π₯|
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PROBLEM SET #4 β Indeterminate Form ***Calculators Not Allowed***
Evaluate the following limits:
1. limxβ0
β3x2 +7x
x= ________________
2. limxββ3
x2 β 9
x + 3= ________________
3. limxββ2
x2 + 4x + 4
x + 2= _______________
4. lim xβ7
49 β x2
x β 7= ________________
5. limxβ0
4x2 + 10x
x= ________________
6. limxββ1
x2 + 3x + 2
x + 1= _______________
7. limxβ1
x10 β 1
x5 β 1= ________________
8. limxββ2
x3 + 5x2 + 6x
x + 2= ____________
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9.ββ limxββ
x
x + 1= ________________
10. limxβ1
x4 β 1
x2 β 1= ________________
11. limxββ2
x3 + 4x2 + 4x
x + 2= ____________
12. limxβ3
x3 β 4x2 + 5x β 6
x β 3= _______
13. limxββ2
x3 + 2x2 β 3x β 6
x + 2= ______
14. limxββ3
x3 + 4x2 + 7x + 12
x + 3= ____
15. limxβ2
x3 β 2x2 + 5x β 10
x β 2= ______
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PROBLEM SET #5 β Infinite Limits ***Calculators Not Allowed***
Use a graphical or number line approach to evaluate the following limits:
1. limxββ1β
x + 8
x + 1= ________
2. limxββ1+
x + 8
x + 1= ________
3. limxββ1
x + 8
x + 1= _________
4. limxβ7β
(x + 7)2
(x β 7)2= ________
5. limxβ7+
(x + 7)2
(x β 7)2= ________
6. limxβ7
(x + 7)2
(x β 7)2= ________
7. limxβ0
x β 4
x= ________
8. limxββ3
2
(x + 3)2= ________
9. limxββ2
x β 4
x2 + 4x + 4= ________
10. limxβ1
7
x3 β 1= ________
11. limxββ3
x β 3
x3 + 6x2 + 9x= ________
12. limxβ3
x + 1
x3 β 4x2 β 13x β 10= _____
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PROBLEM SET #6 β Piecewise Defined Limits ***Calculators Not Allowed***
Evaluate the following limits using the given piecewise functions:
π(π₯) = {cos π₯ , π₯ β€ 0
β3π₯ + 1, 0 < π₯ β€ 2
π₯2 β 4π₯ β 1, π₯ > 2 π(π₯) = {
2 sin(2π₯) , π₯ β€ βπ
2tan(2π₯) , βπ < π₯ β€π
41
2, π₯ >
π
4
β(π₯) = {
3π₯, π₯ β€ 0|cos(π₯)|, 0 < π₯ β€ π
3(π₯ β π) + 1, π₯ > π π(π₯) =
{
π₯2β10π₯
10π₯β100, π₯ β€ 10
log π₯ , 10 < π₯ β€ 1001
βπ₯, π₯ > 100
1. lim
π₯β0π(π₯) = ________________
2. lim
π₯β2π(π₯) = ________________
3. lim
π₯β1π(π₯) = ________________
4. lim
π₯β3π(π₯) = ________________
5. lim
π₯ββππ(π₯) = ________________
6. lim
π₯βπ4
π(π₯) = ________________
7. lim
π₯β0π(π₯) = ________________
8. lim
π₯βπ2
π(π₯) = _________________
9. limπ₯ββ1
β(π₯) = ________________
10. lim
π₯β0β(π₯) = ________________
11. lim
π₯β2πβ(π₯) = ________________
12. lim
π₯βπβ(π₯) = ________________
13. lim
π₯β121π(π₯) = ________________
14. lim
π₯β0π(π₯) = ________________
15. lim
π₯β10π(π₯) = ________________
16. lim
π₯β100π(π₯) = ________________
NJCTL.org
PROBLEM SET #7 β End Behaviors ***Calculators Not Allowed***
Evaluate the following limits:
1. limxββ
β3
x= ________________
2. limxββ
x2 β 9
x + 7= ________________
3. limxββ
x2 + 4x + 4
x2 + 6x + 9= _______________
4. lim xββ
49 β x2
x2 β 16= ________________
5. limxββ
4x2 + x + 5
7x2 + 2x + 3= ________________
6. limxββ
x3
(x + 100)2= ________________
7. limxββ
x + 1x
x= ________________
8. limxββ
2π₯
3π₯= ________________
9. lim xββ
4π₯
3π₯= ________________
10. lim xββ
βx + 2
x + 2= ________________
11. ββ limxββ
sin π₯
x= ________________
12. lim
xββln π₯ = ________________
13. limxββ
x4 β 1
3π₯= ________________
14. limxββ
xπ₯
4π₯= ________________
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15. lim xββ
8 β 4x2 + 3x3 β x
(2 β x)3 = _________
16. lim xβββ
3x2 β 7x3 + 4
14π₯5 + 7π₯3 β 2π₯ + 1 = ______
17. lim xβββ
12x3 β 5x7 + 3x
8π₯2 β 2π₯6 + 5π₯ β 3 = ________
18. lim xβββ
β16x4 + 2x β 7
2π₯2 + 5 = _________
19. lim xβββ
3x + 17x6
β2π₯3 + 11 = _________
20. lim xβββ
βx7 + 4x β 2
5x β 2x2 = _________
21. lim xββ
βx2 + 3
2π₯ β 1 = _________
22. lim xβββ
βx2 + 3
2π₯ β 1 = _________
23. lim xββ
β4x4 + 2
3π₯2 + 5 = _________
24. lim xβββ
β4x4 + 2
3π₯2 + 5 = _________
25. lim xββ
βx4 + 2 β x2 = _________
26. lim xββ
βx4 + 2x β x2 =_________
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PROBLEM SET #8 β Trig Limits ***Calculators Not Allowed***
Evaluate the following limits:
1. limxβ0
sin 6π₯
3π₯= ________________
2. limxβ0
7π₯
cos(7π₯)= ________________
3. limxβ0
tan π₯
sin π₯= _______________
4. lim xβ0
sin 5π₯
sin 7π₯= ________________
5. limxβ0
2 β 2cos(π₯)
π₯= _____________
6. limxβ0
4β 4 cos2 π₯
sin2 π₯ ________________
7. limxβ0
sin2 2π₯
4π₯2= ________________
8. limxβ0
tan2(4 π₯)
π₯2= ________________
9. limxβ0
π₯ csc π₯ = _________________
10. lim xβ0
π₯2
sin(π₯) β 1= _____________
11. limxβ0
sin2 π₯
6π₯= ________________
12. limxβ0
π₯
tan π₯= _________________
13. limxβ0
1 β sec π₯
π₯= _______________
14. limxβ0
4π₯
sin π₯= ________________
15. limxβ0
π₯ + sin π₯
sin π₯= _______________
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PROBLEM SET #9 β Difference Quotient ***Calculators Not Allowed**
Use the difference quotient to answer the following questions.
1. Calculate the slope of the tangent line
to π¦ = π₯2 β 5 at π₯ = 0.
2. Calculate the slope of the tangent line
to π¦ = 2π₯2 β 4π₯ + 4 at π₯ = β1.
3. Calculate the slope of the tangent line
to π¦ = 3π₯2 β 4π₯ + 5 at π₯ = 1.
4. Calculate the slope of the tangent line
to π¦ = π₯3 at any value x.
5. Calculate the slope of the tangent line
to π¦ = 2π₯3 + 1 at π₯ = β2.
6. Calculate the slope of the tangent line
to π¦ =1
π₯ at π₯ = 1.
7. Calculate the slope of the tangent line
to π¦ = β2
π₯ at π₯ = 1.
8. Calculate the slope of the tangent line
to π¦ = 10 at any value x.
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9. Calculate the slope of the tangent line
to π¦ =1
π₯+4 at π₯ = 2.
10. Calculate the slope of the tangent
line to π¦ =π₯
π₯β2 at π₯ = 3.
11. Calculate the slope of the tangent
line to π¦ = βπ₯ at any value x.
12. Calculate the slope of the tangent
line to π¦ = βπ₯ + 3 at π₯ = 6.
13. Calculate the slope of the tangent
line to π¦ = π πππ₯ at π₯ = 0.
14. ** Calculate the slope of the tangent
line to π¦ = πππ₯ at π₯ = 7.
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Limits and Continuity- Answer Keys Problem Set #1 β Tangent Lines
1. 2 2. -4
3. π¦ β 8 = 4(π₯ β1
2) or π¦ =
4π₯ + 6 4. 5 5. 1 6. π¦ = π₯ 7. 0 8. 2/3 9. π¦ β 8 = 12(π₯ β 3) or π¦ =12π₯ β 28
10. 1/4
Problem Set #2β Limits (Graphs)
1. 0 2. DNE 3. 1.25 4. 1 5. 4 or β 6. 0 7. 1 8. -0.75
Problem Set #3 β Computing Limits
1. 4 2. 0 3. -6 4. 0 5. 1 6. ln(3)+1 7. 3 8. 0 9. DNE 10. 0 11. 30
12. 0 13. DNE 14. 2 15. DNE
Problem Set #4 β Indeterminate Form
1. 7 2. -6 3. 0 4. -14 5. 10 6. 1 7. 2 8. -2 9. 1 10. 2 11. 0 12. 8 13. 1 14. 10 15. 9
Problem Set #5 β Infinite Limits
1. -β 2. +β 3. DNE 4. +β 5. +β 6. +β 7. DNE 8. +β 9. -β 10. DNE 11. +β 12. DNE
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Problem Set #6β Piecewise Limits
1. 1 2. -5 3. -2 4. -4 5. 0 6. DNE 7. 0 8. 1/2 9. 1/3 10. 1 11. 3π + 1 12. 1 13. 1/11 14. 0 15. 1 16. DNE
Problem Set #7 β End Behaviors
1. 0 2. β 3. 1 4. -1 5. 4/7 6. β 7. 1 8. 0 9. β 10. 0 11. 0 12. β 13. 0 14. β 15. -3 16. 0 17. ββ 18. ββ 19. β 20. ββ
21. 1/2 22. -1/2 23. 2/3 24. 2/3 25. 0 26. 0
Problem Set #8 β Trig Lines
1. 2 2. 0 3. 1 4. 5/7 5. 0 6. 4 7. 1 8. 16 9. 1 10. 0 11. 0 12. 1 13. 0 14. 4 15. 2
Problem Set #9 β Difference Quotient
1. 0 2. -8 3. 2 4. 3x2 5. 24 6. -1 7. 2 8. 0 9. -1/36 10. -2
11. 1
2βπ₯
12. 1/6 13. 1 14. 1/7
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