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NJCTL.org LIMITS UNIT PROBLEM SETS PROBLEM SET #1 – Tangent Lines ***Calculators Not Allowed*** 1. Given: () = 2 βˆ’ 7 Find the slope of the tangent line at 0 =3 2. Given: () = βˆ’4 βˆ’ 2 Find the slope of the secant line between 1 = βˆ’2 and 2 =3 3. Given: () = 4 2 +7 Find the equation of the tangent line at 0 = 1 2 4. Given: () = βˆ’2 2 βˆ’ 3 Find the slope of the tangent line at 0 = βˆ’2 5. Given: () = 3 +8 Find the slope of the secant line between 1 =0 and 2 =1

LIMITS UNIT PROBLEM SETS PROBLEM SET #1 – Tangent Lines

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Page 1: LIMITS UNIT PROBLEM SETS PROBLEM SET #1 – Tangent Lines

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LIMITS UNIT PROBLEM SETS

PROBLEM SET #1 – Tangent Lines ***Calculators Not Allowed***

1. Given: 𝑓(π‘₯) = 2π‘₯ βˆ’ 7 Find the slope of the tangent line at π‘₯0 = 3

2. Given: 𝑓(π‘₯) = βˆ’4π‘₯ βˆ’ 2 Find the slope of the secant line between π‘₯1 = βˆ’2

and π‘₯2 = 3

3. Given: 𝑓(π‘₯) = 4π‘₯2 + 7 Find the equation of the tangent line at π‘₯0 =1

2

4. Given: 𝑓(π‘₯) = βˆ’2π‘₯2 βˆ’ 3π‘₯ Find the slope of the tangent line at π‘₯0 = βˆ’2

5. Given: 𝑓(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: 𝑓(π‘₯) = 2π‘₯3 + π‘₯ Find the equation of the tangent line at π‘₯0 = 0

7. Given: 𝑓(π‘₯) = 7 Find the slope of the tangent line at π‘₯0 = 1

8. Given: 𝑓(π‘₯) =2

3π‘₯ + 4 Find the slope of the secant line between π‘₯1 = 6 and

π‘₯2 = 9

9. Given: 𝑓(π‘₯) = 2π‘₯2 βˆ’ 10 Find the equation of the tangent line at π‘₯0 = 3

10. Given: 𝑓(π‘₯) = βˆ’1

π‘₯ Find the slope of the tangent line at π‘₯0 = 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

π‘₯β†’0𝑓(π‘₯) = ________________

2. limπ‘₯β†’2

𝑓(π‘₯) = ________________

3. limπ‘₯β†’βˆ’1βˆ’

𝑓(π‘₯) = ________________

4. limπ‘₯β†’βˆ’1+

𝑓(π‘₯) = ________________

Use for problems #1-4

5. limπ‘₯β†’βˆ’1βˆ’

𝑓(π‘₯) = ________________

6. limπ‘₯β†’βˆ’1+

𝑓(π‘₯) = ________________

7. limπ‘₯β†’3βˆ’

𝑓(π‘₯) = ________________

8. limπ‘₯β†’3+

𝑓(π‘₯) = ________________

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. limπ‘₯β†’

πœ‹2

3sin π‘₯

2. limπ‘₯β†’

12

(βˆ’4π‘₯ + 2)

3. limπ‘₯β†’3

(βˆ’3π‘₯2 + 7π‘₯)

4. lim π‘₯β†’0

π‘₯(3π‘₯2 + 7)

5. limπ‘₯→𝑒

ln|π‘₯|

6. limπ‘₯→𝑒

ln|3π‘₯|

7. limπ‘₯β†’3

|π‘₯4 βˆ’ 2π‘₯3 βˆ’ 30|

8. limπ‘₯β†’3+

√(π‘₯2 βˆ’ 9)

9. limπ‘₯β†’4βˆ’

√(π‘₯2 βˆ’ 16)

10. limπ‘₯β†’3

(π‘₯ + 2)(π‘₯ βˆ’ 3)

11. limπ‘₯β†’βˆ’2

(βˆ’3π‘₯3 + 4π‘₯2 βˆ’ 10)

12. limπ‘₯β†’

πœ‹2

cot π‘₯

13. limπ‘₯β†’βˆ’5

√(π‘₯ + 4)

14. limπ‘₯β†’πœ‹

2 cos(2π‘₯)

15. limπ‘₯β†’0

ln|2π‘₯|

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PROBLEM SET #4 – Indeterminate Form ***Calculators Not Allowed***

Evaluate the following limits:

1. limx→0

βˆ’3x2 +7x

x= ________________

2. limxβ†’βˆ’3

x2 βˆ’ 9

x + 3= ________________

3. limxβ†’βˆ’2

x2 + 4x + 4

x + 2= _______________

4. lim x→7

49 βˆ’ x2

x βˆ’ 7= ________________

5. limx→0

4x2 + 10x

x= ________________

6. limxβ†’βˆ’1

x2 + 3x + 2

x + 1= _______________

7. limx→1

x10 βˆ’ 1

x5 βˆ’ 1= ________________

8. limxβ†’βˆ’2

x3 + 5x2 + 6x

x + 2= ____________

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9.βˆ—βˆ— limxβ†’βˆž

x

x + 1= ________________

10. limx→1

x4 βˆ’ 1

x2 βˆ’ 1= ________________

11. limxβ†’βˆ’2

x3 + 4x2 + 4x

x + 2= ____________

12. limx→3

x3 βˆ’ 4x2 + 5x βˆ’ 6

x βˆ’ 3= _______

13. limxβ†’βˆ’2

x3 + 2x2 βˆ’ 3x βˆ’ 6

x + 2= ______

14. limxβ†’βˆ’3

x3 + 4x2 + 7x + 12

x + 3= ____

15. limx→2

x3 βˆ’ 2x2 + 5x βˆ’ 10

x βˆ’ 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. limxβ†’βˆ’1βˆ’

x + 8

x + 1= ________

2. limxβ†’βˆ’1+

x + 8

x + 1= ________

3. limxβ†’βˆ’1

x + 8

x + 1= _________

4. limxβ†’7βˆ’

(x + 7)2

(x βˆ’ 7)2= ________

5. limx→7+

(x + 7)2

(x βˆ’ 7)2= ________

6. limx→7

(x + 7)2

(x βˆ’ 7)2= ________

7. limx→0

x βˆ’ 4

x= ________

8. limxβ†’βˆ’3

2

(x + 3)2= ________

9. limxβ†’βˆ’2

x βˆ’ 4

x2 + 4x + 4= ________

10. limx→1

7

x3 βˆ’ 1= ________

11. limxβ†’βˆ’3

x βˆ’ 3

x3 + 6x2 + 9x= ________

12. limx→3

x + 1

x3 βˆ’ 4x2 βˆ’ 13x βˆ’ 10= _____

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PROBLEM SET #6 – Piecewise Defined Limits ***Calculators Not Allowed***

Evaluate the following limits using the given piecewise functions:

𝑓(π‘₯) = {cos π‘₯ , π‘₯ ≀ 0

βˆ’3π‘₯ + 1, 0 < π‘₯ ≀ 2

π‘₯2 βˆ’ 4π‘₯ βˆ’ 1, π‘₯ > 2 𝑔(π‘₯) = {

2 sin(2π‘₯) , π‘₯ ≀ βˆ’πœ‹

2tan(2π‘₯) , βˆ’πœ‹ < π‘₯ β‰€πœ‹

41

2, π‘₯ >

πœ‹

4

β„Ž(π‘₯) = {

3π‘₯, π‘₯ ≀ 0|cos(π‘₯)|, 0 < π‘₯ ≀ πœ‹

3(π‘₯ βˆ’ πœ‹) + 1, π‘₯ > πœ‹ π‘˜(π‘₯) =

{

π‘₯2βˆ’10π‘₯

10π‘₯βˆ’100, π‘₯ ≀ 10

log π‘₯ , 10 < π‘₯ ≀ 1001

√π‘₯, π‘₯ > 100

1. lim

π‘₯β†’0𝑓(π‘₯) = ________________

2. lim

π‘₯β†’2𝑓(π‘₯) = ________________

3. lim

π‘₯β†’1𝑓(π‘₯) = ________________

4. lim

π‘₯β†’3𝑓(π‘₯) = ________________

5. lim

π‘₯β†’βˆ’πœ‹π‘”(π‘₯) = ________________

6. lim

π‘₯β†’πœ‹4

𝑔(π‘₯) = ________________

7. lim

π‘₯β†’0𝑔(π‘₯) = ________________

8. lim

π‘₯β†’πœ‹2

𝑔(π‘₯) = _________________

9. limπ‘₯β†’βˆ’1

β„Ž(π‘₯) = ________________

10. lim

π‘₯β†’0β„Ž(π‘₯) = ________________

11. lim

π‘₯β†’2πœ‹β„Ž(π‘₯) = ________________

12. lim

π‘₯β†’πœ‹β„Ž(π‘₯) = ________________

13. lim

π‘₯β†’121π‘˜(π‘₯) = ________________

14. lim

π‘₯β†’0π‘˜(π‘₯) = ________________

15. lim

π‘₯β†’10π‘˜(π‘₯) = ________________

16. lim

π‘₯β†’100π‘˜(π‘₯) = ________________

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PROBLEM SET #7 – End Behaviors ***Calculators Not Allowed***

Evaluate the following limits:

1. limxβ†’βˆž

βˆ’3

x= ________________

2. limxβ†’βˆž

x2 βˆ’ 9

x + 7= ________________

3. limxβ†’βˆž

x2 + 4x + 4

x2 + 6x + 9= _______________

4. lim xβ†’βˆž

49 βˆ’ x2

x2 βˆ’ 16= ________________

5. limxβ†’βˆž

4x2 + x + 5

7x2 + 2x + 3= ________________

6. limxβ†’βˆž

x3

(x + 100)2= ________________

7. limxβ†’βˆž

x + 1x

x= ________________

8. limxβ†’βˆž

2π‘₯

3π‘₯= ________________

9. lim xβ†’βˆž

4π‘₯

3π‘₯= ________________

10. lim xβ†’βˆž

√x + 2

x + 2= ________________

11. βˆ—βˆ— limxβ†’βˆž

sin π‘₯

x= ________________

12. lim

xβ†’βˆžln π‘₯ = ________________

13. limxβ†’βˆž

x4 βˆ’ 1

3π‘₯= ________________

14. limxβ†’βˆž

xπ‘₯

4π‘₯= ________________

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15. lim xβ†’βˆž

8 βˆ’ 4x2 + 3x3 βˆ’ x

(2 βˆ’ x)3 = _________

16. lim xβ†’βˆ’βˆž

3x2 βˆ’ 7x3 + 4

14π‘₯5 + 7π‘₯3 βˆ’ 2π‘₯ + 1 = ______

17. lim xβ†’βˆ’βˆž

12x3 βˆ’ 5x7 + 3x

8π‘₯2 βˆ’ 2π‘₯6 + 5π‘₯ βˆ’ 3 = ________

18. lim xβ†’βˆ’βˆž

βˆ’16x4 + 2x βˆ’ 7

2π‘₯2 + 5 = _________

19. lim xβ†’βˆ’βˆž

3x + 17x6

βˆ’2π‘₯3 + 11 = _________

20. lim xβ†’βˆ’βˆž

βˆ’x7 + 4x βˆ’ 2

5x βˆ’ 2x2 = _________

21. lim xβ†’βˆž

√x2 + 3

2π‘₯ βˆ’ 1 = _________

22. lim xβ†’βˆ’βˆž

√x2 + 3

2π‘₯ βˆ’ 1 = _________

23. lim xβ†’βˆž

√4x4 + 2

3π‘₯2 + 5 = _________

24. lim xβ†’βˆ’βˆž

√4x4 + 2

3π‘₯2 + 5 = _________

25. lim xβ†’βˆž

√x4 + 2 βˆ’ x2 = _________

26. lim xβ†’βˆž

√x4 + 2x βˆ’ x2 =_________

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PROBLEM SET #8 – Trig Limits ***Calculators Not Allowed***

Evaluate the following limits:

1. limx→0

sin 6π‘₯

3π‘₯= ________________

2. limx→0

7π‘₯

cos(7π‘₯)= ________________

3. limx→0

tan π‘₯

sin π‘₯= _______________

4. lim x→0

sin 5π‘₯

sin 7π‘₯= ________________

5. limx→0

2 βˆ’ 2cos(π‘₯)

π‘₯= _____________

6. limx→0

4βˆ’ 4 cos2 π‘₯

sin2 π‘₯ ________________

7. limx→0

sin2 2π‘₯

4π‘₯2= ________________

8. limx→0

tan2(4 π‘₯)

π‘₯2= ________________

9. limx→0

π‘₯ csc π‘₯ = _________________

10. lim x→0

π‘₯2

sin(π‘₯) βˆ’ 1= _____________

11. limx→0

sin2 π‘₯

6π‘₯= ________________

12. limx→0

π‘₯

tan π‘₯= _________________

13. limx→0

1 βˆ’ sec π‘₯

π‘₯= _______________

14. limx→0

4π‘₯

sin π‘₯= ________________

15. limx→0

π‘₯ + sin π‘₯

sin π‘₯= _______________

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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 = π‘₯2 βˆ’ 5 at π‘₯ = 0.

2. Calculate the slope of the tangent line

to 𝑦 = 2π‘₯2 βˆ’ 4π‘₯ + 4 at π‘₯ = βˆ’1.

3. Calculate the slope of the tangent line

to 𝑦 = 3π‘₯2 βˆ’ 4π‘₯ + 5 at π‘₯ = 1.

4. Calculate the slope of the tangent line

to 𝑦 = π‘₯3 at any value x.

5. Calculate the slope of the tangent line

to 𝑦 = 2π‘₯3 + 1 at π‘₯ = βˆ’2.

6. Calculate the slope of the tangent line

to 𝑦 =1

π‘₯ at π‘₯ = 1.

7. Calculate the slope of the tangent line

to 𝑦 = βˆ’2

π‘₯ at π‘₯ = 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

π‘₯+4 at π‘₯ = 2.

10. Calculate the slope of the tangent

line to 𝑦 =π‘₯

π‘₯βˆ’2 at π‘₯ = 3.

11. Calculate the slope of the tangent

line to 𝑦 = √π‘₯ at any value x.

12. Calculate the slope of the tangent

line to 𝑦 = √π‘₯ + 3 at π‘₯ = 6.

13. Calculate the slope of the tangent

line to 𝑦 = 𝑠𝑖𝑛π‘₯ at π‘₯ = 0.

14. ** Calculate the slope of the tangent

line to 𝑦 = 𝑙𝑛π‘₯ at π‘₯ = 7.

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Limits and Continuity- Answer Keys Problem Set #1 – Tangent Lines

1. 2 2. -4

3. 𝑦 βˆ’ 8 = 4(π‘₯ βˆ’1

2) or 𝑦 =

4π‘₯ + 6 4. 5 5. 1 6. 𝑦 = π‘₯ 7. 0 8. 2/3 9. 𝑦 βˆ’ 8 = 12(π‘₯ βˆ’ 3) or 𝑦 =12π‘₯ βˆ’ 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œ‹ + 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. βˆ’βˆž 18. βˆ’βˆž 19. ∞ 20. βˆ’βˆž

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√π‘₯

12. 1/6 13. 1 14. 1/7