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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

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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

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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

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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|

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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= ____________

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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= ______

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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= _____

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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) = ________________

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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= ________________

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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 =_________

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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= _______________

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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.

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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.

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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

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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