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NJCTL.org
LIMITS UNIT PROBLEM SETS
PROBLEM SET #1 – Tangent Lines ***Calculators Not Allowed***
1. Given: 𝑓(𝑥) = 2𝑥 − 7 Find the slope of the tangent line at 𝑥0 = 3
2. Given: 𝑓(𝑥) = −4𝑥 − 2 Find the slope of the secant line between 𝑥1 = −2
and 𝑥2 = 3
3. Given: 𝑓(𝑥) = 4𝑥2 + 7 Find the equation of the tangent line at 𝑥0 =1
2
4. Given: 𝑓(𝑥) = −2𝑥2 − 3𝑥 Find the slope of the tangent line at 𝑥0 = −2
5. Given: 𝑓(𝑥) = 𝑥3 + 8 Find the slope of the secant line between 𝑥1 = 0 and 𝑥2 = 1
NJCTL.org
6. Given: 𝑓(𝑥) = 2𝑥3 + 𝑥 Find the equation of the tangent line at 𝑥0 = 0
7. Given: 𝑓(𝑥) = 7 Find the slope of the tangent line at 𝑥0 = 1
8. Given: 𝑓(𝑥) =2
3𝑥 + 4 Find the slope of the secant line between 𝑥1 = 6 and
𝑥2 = 9
9. Given: 𝑓(𝑥) = 2𝑥2 − 10 Find the equation of the tangent line at 𝑥0 = 3
10. Given: 𝑓(𝑥) = −1
𝑥 Find the slope of the tangent line at 𝑥0 = 2
NJCTL.org
PROBLEM SET #2 – Limits (Graphs) ***Calculators Not Allowed***
For problems #1-8, find the limit of the function at the given point:
1. lim
𝑥→0𝑓(𝑥) = ________________
2. lim𝑥→2
𝑓(𝑥) = ________________
3. lim𝑥→−1−
𝑓(𝑥) = ________________
4. lim𝑥→−1+
𝑓(𝑥) = ________________
Use for problems #1-4
5. lim𝑥→−1−
𝑓(𝑥) = ________________
6. lim𝑥→−1+
𝑓(𝑥) = ________________
7. lim𝑥→3−
𝑓(𝑥) = ________________
8. lim𝑥→3+
𝑓(𝑥) = ________________
Use for problems #5-8
NJCTL.org
PROBLEM SET #3 – Computing Limits ***Calculators Not Allowed***
For the following, find the limit of the function at the given point:
1. lim𝑥→
𝜋2
3sin 𝑥
2. lim𝑥→
12
(−4𝑥 + 2)
3. lim𝑥→3
(−3𝑥2 + 7𝑥)
4. lim 𝑥→0
𝑥(3𝑥2 + 7)
5. lim𝑥→𝑒
ln|𝑥|
6. lim𝑥→𝑒
ln|3𝑥|
7. lim𝑥→3
|𝑥4 − 2𝑥3 − 30|
8. lim𝑥→3+
√(𝑥2 − 9)
9. lim𝑥→4−
√(𝑥2 − 16)
10. lim𝑥→3
(𝑥 + 2)(𝑥 − 3)
11. lim𝑥→−2
(−3𝑥3 + 4𝑥2 − 10)
12. lim𝑥→
𝜋2
cot 𝑥
13. lim𝑥→−5
√(𝑥 + 4)
14. lim𝑥→𝜋
2 cos(2𝑥)
15. lim𝑥→0
ln|2𝑥|
NJCTL.org
PROBLEM SET #4 – Indeterminate Form ***Calculators Not Allowed***
Evaluate the following limits:
1. limx→0
−3x2 +7x
x= ________________
2. limx→−3
x2 − 9
x + 3= ________________
3. limx→−2
x2 + 4x + 4
x + 2= _______________
4. lim x→7
49 − x2
x − 7= ________________
5. limx→0
4x2 + 10x
x= ________________
6. limx→−1
x2 + 3x + 2
x + 1= _______________
7. limx→1
x10 − 1
x5 − 1= ________________
8. limx→−2
x3 + 5x2 + 6x
x + 2= ____________
NJCTL.org
9.∗∗ limx→∞
x
x + 1= ________________
10. limx→1
x4 − 1
x2 − 1= ________________
11. limx→−2
x3 + 4x2 + 4x
x + 2= ____________
12. limx→3
x3 − 4x2 + 5x − 6
x − 3= _______
13. limx→−2
x3 + 2x2 − 3x − 6
x + 2= ______
14. limx→−3
x3 + 4x2 + 7x + 12
x + 3= ____
15. limx→2
x3 − 2x2 + 5x − 10
x − 2= ______
NJCTL.org
PROBLEM SET #5 – Infinite Limits ***Calculators Not Allowed***
Use a graphical or number line approach to evaluate the following limits:
1. limx→−1−
x + 8
x + 1= ________
2. limx→−1+
x + 8
x + 1= ________
3. limx→−1
x + 8
x + 1= _________
4. limx→7−
(x + 7)2
(x − 7)2= ________
5. limx→7+
(x + 7)2
(x − 7)2= ________
6. limx→7
(x + 7)2
(x − 7)2= ________
7. limx→0
x − 4
x= ________
8. limx→−3
2
(x + 3)2= ________
9. limx→−2
x − 4
x2 + 4x + 4= ________
10. limx→1
7
x3 − 1= ________
11. limx→−3
x − 3
x3 + 6x2 + 9x= ________
12. limx→3
x + 1
x3 − 4x2 − 13x − 10= _____
NJCTL.org
PROBLEM SET #6 – Piecewise Defined Limits ***Calculators Not Allowed***
Evaluate the following limits using the given piecewise functions:
𝑓(𝑥) = {cos 𝑥 , 𝑥 ≤ 0
−3𝑥 + 1, 0 < 𝑥 ≤ 2
𝑥2 − 4𝑥 − 1, 𝑥 > 2 𝑔(𝑥) = {
2 sin(2𝑥) , 𝑥 ≤ −𝜋
2tan(2𝑥) , −𝜋 < 𝑥 ≤𝜋
41
2, 𝑥 >
𝜋
4
ℎ(𝑥) = {
3𝑥, 𝑥 ≤ 0|cos(𝑥)|, 0 < 𝑥 ≤ 𝜋
3(𝑥 − 𝜋) + 1, 𝑥 > 𝜋 𝑘(𝑥) =
{
𝑥2−10𝑥
10𝑥−100, 𝑥 ≤ 10
log 𝑥 , 10 < 𝑥 ≤ 1001
√𝑥, 𝑥 > 100
1. lim
𝑥→0𝑓(𝑥) = ________________
2. lim
𝑥→2𝑓(𝑥) = ________________
3. lim
𝑥→1𝑓(𝑥) = ________________
4. lim
𝑥→3𝑓(𝑥) = ________________
5. lim
𝑥→−𝜋𝑔(𝑥) = ________________
6. lim
𝑥→𝜋4
𝑔(𝑥) = ________________
7. lim
𝑥→0𝑔(𝑥) = ________________
8. lim
𝑥→𝜋2
𝑔(𝑥) = _________________
9. lim𝑥→−1
ℎ(𝑥) = ________________
10. lim
𝑥→0ℎ(𝑥) = ________________
11. lim
𝑥→2𝜋ℎ(𝑥) = ________________
12. lim
𝑥→𝜋ℎ(𝑥) = ________________
13. lim
𝑥→121𝑘(𝑥) = ________________
14. lim
𝑥→0𝑘(𝑥) = ________________
15. lim
𝑥→10𝑘(𝑥) = ________________
16. lim
𝑥→100𝑘(𝑥) = ________________
NJCTL.org
PROBLEM SET #7 – End Behaviors ***Calculators Not Allowed***
Evaluate the following limits:
1. limx→∞
−3
x= ________________
2. limx→∞
x2 − 9
x + 7= ________________
3. limx→∞
x2 + 4x + 4
x2 + 6x + 9= _______________
4. lim x→∞
49 − x2
x2 − 16= ________________
5. limx→∞
4x2 + x + 5
7x2 + 2x + 3= ________________
6. limx→∞
x3
(x + 100)2= ________________
7. limx→∞
x + 1x
x= ________________
8. limx→∞
2𝑥
3𝑥= ________________
9. lim x→∞
4𝑥
3𝑥= ________________
10. lim x→∞
√x + 2
x + 2= ________________
11. ∗∗ limx→∞
sin 𝑥
x= ________________
12. lim
x→∞ln 𝑥 = ________________
13. limx→∞
x4 − 1
3𝑥= ________________
14. limx→∞
x𝑥
4𝑥= ________________
NJCTL.org
15. lim x→∞
8 − 4x2 + 3x3 − x
(2 − x)3 = _________
16. lim x→−∞
3x2 − 7x3 + 4
14𝑥5 + 7𝑥3 − 2𝑥 + 1 = ______
17. lim x→−∞
12x3 − 5x7 + 3x
8𝑥2 − 2𝑥6 + 5𝑥 − 3 = ________
18. lim x→−∞
−16x4 + 2x − 7
2𝑥2 + 5 = _________
19. lim x→−∞
3x + 17x6
−2𝑥3 + 11 = _________
20. lim x→−∞
−x7 + 4x − 2
5x − 2x2 = _________
21. lim x→∞
√x2 + 3
2𝑥 − 1 = _________
22. lim x→−∞
√x2 + 3
2𝑥 − 1 = _________
23. lim x→∞
√4x4 + 2
3𝑥2 + 5 = _________
24. lim x→−∞
√4x4 + 2
3𝑥2 + 5 = _________
25. lim x→∞
√x4 + 2 − x2 = _________
26. lim x→∞
√x4 + 2x − x2 =_________
NJCTL.org
PROBLEM SET #8 – Trig Limits ***Calculators Not Allowed***
Evaluate the following limits:
1. limx→0
sin 6𝑥
3𝑥= ________________
2. limx→0
7𝑥
cos(7𝑥)= ________________
3. limx→0
tan 𝑥
sin 𝑥= _______________
4. lim x→0
sin 5𝑥
sin 7𝑥= ________________
5. limx→0
2 − 2cos(𝑥)
𝑥= _____________
6. limx→0
4− 4 cos2 𝑥
sin2 𝑥 ________________
7. limx→0
sin2 2𝑥
4𝑥2= ________________
8. limx→0
tan2(4 𝑥)
𝑥2= ________________
9. limx→0
𝑥 csc 𝑥 = _________________
10. lim x→0
𝑥2
sin(𝑥) − 1= _____________
11. limx→0
sin2 𝑥
6𝑥= ________________
12. limx→0
𝑥
tan 𝑥= _________________
13. limx→0
1 − sec 𝑥
𝑥= _______________
14. limx→0
4𝑥
sin 𝑥= ________________
15. limx→0
𝑥 + sin 𝑥
sin 𝑥= _______________
NJCTL.org
PROBLEM SET #9 – Difference Quotient ***Calculators Not Allowed**
Use the difference quotient to answer the following questions.
1. Calculate the slope of the tangent line
to 𝑦 = 𝑥2 − 5 at 𝑥 = 0.
2. Calculate the slope of the tangent line
to 𝑦 = 2𝑥2 − 4𝑥 + 4 at 𝑥 = −1.
3. Calculate the slope of the tangent line
to 𝑦 = 3𝑥2 − 4𝑥 + 5 at 𝑥 = 1.
4. Calculate the slope of the tangent line
to 𝑦 = 𝑥3 at any value x.
5. Calculate the slope of the tangent line
to 𝑦 = 2𝑥3 + 1 at 𝑥 = −2.
6. Calculate the slope of the tangent line
to 𝑦 =1
𝑥 at 𝑥 = 1.
7. Calculate the slope of the tangent line
to 𝑦 = −2
𝑥 at 𝑥 = 1.
8. Calculate the slope of the tangent line
to 𝑦 = 10 at any value x.
NJCTL.org
9. Calculate the slope of the tangent line
to 𝑦 =1
𝑥+4 at 𝑥 = 2.
10. Calculate the slope of the tangent
line to 𝑦 =𝑥
𝑥−2 at 𝑥 = 3.
11. Calculate the slope of the tangent
line to 𝑦 = √𝑥 at any value x.
12. Calculate the slope of the tangent
line to 𝑦 = √𝑥 + 3 at 𝑥 = 6.
13. Calculate the slope of the tangent
line to 𝑦 = 𝑠𝑖𝑛𝑥 at 𝑥 = 0.
14. ** Calculate the slope of the tangent
line to 𝑦 = 𝑙𝑛𝑥 at 𝑥 = 7.
NJCTL.org
Limits and Continuity- Answer Keys Problem Set #1 – Tangent Lines
1. 2 2. -4
3. 𝑦 − 8 = 4(𝑥 −1
2) or 𝑦 =
4𝑥 + 6 4. 5 5. 1 6. 𝑦 = 𝑥 7. 0 8. 2/3 9. 𝑦 − 8 = 12(𝑥 − 3) or 𝑦 =12𝑥 − 28
10. 1/4
Problem Set #2– Limits (Graphs)
1. 0 2. DNE 3. 1.25 4. 1 5. 4 or ∞ 6. 0 7. 1 8. -0.75
Problem Set #3 – 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 – 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 – Infinite Limits
1. -∞ 2. +∞ 3. DNE 4. +∞ 5. +∞ 6. +∞ 7. DNE 8. +∞ 9. -∞ 10. DNE 11. +∞ 12. DNE
NJCTL.org
Problem Set #6– 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 – End Behaviors
1. 0 2. ∞ 3. 1 4. -1 5. 4/7 6. ∞ 7. 1 8. 0 9. ∞ 10. 0 11. 0 12. ∞ 13. 0 14. ∞ 15. -3 16. 0 17. −∞ 18. −∞ 19. ∞ 20. −∞
21. 1/2 22. -1/2 23. 2/3 24. 2/3 25. 0 26. 0
Problem Set #8 – 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 – 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
2√𝑥
12. 1/6 13. 1 14. 1/7