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

LIMITS UNIT PROBLEM SETS

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

1. Given: () = 2 7 Find the slope of the tangent line at 0 = 3

2. Given: () = 4 2 Find the slope of the secant line between 1 = 2

and 2 = 3

3. Given: () = 42 + 7 Find the equation of the tangent line at 0 =1

2

4. Given: () = 22 3 Find the slope of the tangent line at 0 = 2

5. Given: () = 3 + 8 Find the slope of the secant line between 1 = 0 and 2 = 1

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6. Given: () = 23 + Find the equation of the tangent line at 0 = 0

7. Given: () = 7 Find the slope of the tangent line at 0 = 1

8. Given: () =2

3 + 4 Find the slope of the secant line between 1 = 6 and

2 = 9

9. Given: () = 22 10 Find the equation of the tangent line at 0 = 3

10. Given: () = 1

Find the slope of the tangent line at 0 = 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

0() = ________________

2. lim2

() = ________________

3. lim1

() = ________________

4. lim1+

() = ________________

Use for problems #1-4

5. lim1

() = ________________

6. lim1+

() = ________________

7. lim3

() = ________________

8. lim3+

() = ________________

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. lim

2

3sin

2. lim

12

(4 + 2)

3. lim3

(32 + 7)

4. lim 0

(32 + 7)

5. lim

ln||

6. lim

ln|3|

7. lim3

|4 23 30|

8. lim3+

(2 9)

9. lim4

(2 16)

10. lim3

( + 2)( 3)

11. lim2

(33 + 42 10)

12. lim

2

cot

13. lim5

( + 4)

14. lim

2 cos(2)

15. lim0

ln|2|

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

Evaluate the following limits:

1. limx0

3x2 +7x

x= ________________

2. limx3

x2 9

x + 3= ________________

3. limx2

x2 + 4x + 4

x + 2= _______________

4. lim x7

49 x2

x 7= ________________

5. limx0

4x2 + 10x

x= ________________

6. limx1

x2 + 3x + 2

x + 1= _______________

7. limx1

x10 1

x5 1= ________________

8. limx2

x3 + 5x2 + 6x

x + 2= ____________

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9. limx

x

x + 1= ________________

10. limx1

x4 1

x2 1= ________________

11. limx2

x3 + 4x2 + 4x

x + 2= ____________

12. limx3

x3 4x2 + 5x 6

x 3= _______

13. limx2

x3 + 2x2 3x 6

x + 2= ______

14. limx3

x3 + 4x2 + 7x + 12

x + 3= ____

15. limx2

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. limx1

x + 8

x + 1= ________

2. limx1+

x + 8

x + 1= ________

3. limx1

x + 8

x + 1= _________

4. limx7

(x + 7)2

(x 7)2= ________

5. limx7+

(x + 7)2

(x 7)2= ________

6. limx7

(x + 7)2

(x 7)2= ________

7. limx0

x 4

x= ________

8. limx3

2

(x + 3)2= ________

9. limx2

x 4

x2 + 4x + 4= ________

10. limx1

7

x3 1= ________

11. limx3

x 3

x3 + 6x2 + 9x= ________

12. limx3

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:

() = {cos , 0

3 + 1, 0 < 2

2 4 1, > 2 () = {

2 sin(2) ,

2tan(2) , <

41

2, >

4

() = {

3, 0|cos()|, 0 <

3( ) + 1, > () =

{

210

10100, 10

log , 10 < 1001

, > 100

1. lim

0() = ________________

2. lim

2() = ________________

3. lim

1() = ________________

4. lim

3() = ________________

5. lim

() = ________________

6. lim

4

() = ________________

7. lim

0() = ________________

8. lim

2

() = _________________

9. lim1

() = ________________

10. lim

0() = ________________

11. lim

2() = ________________

12. lim

() = ________________

13. lim

121() = ________________

14. lim

0() = ________________

15. lim

10() = ________________

16. lim

100() = ________________

• 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

2

3= ________________

9. lim x

4

3= ________________

10. lim x

x + 2

x + 2= ________________

11. limx

sin

x= ________________

12. lim

xln = ________________

13. limx

x4 1

3= ________________

14. limx

x

4= ________________

• NJCTL.org

15. lim x

8 4x2 + 3x3 x

(2 x)3 = _________

16. lim x

3x2 7x3 + 4

145 + 73 2 + 1 = ______

17. lim x

12x3 5x7 + 3x

82 26 + 5 3 = ________

18. lim x

16x4 + 2x 7

22 + 5 = _________

19. lim x

3x + 17x6

23 + 11 = _________

20. lim x

x7 + 4x 2

5x 2x2 = _________

21. lim x

x2 + 3

2 1 = _________

22. lim x

x2 + 3

2 1 = _________

23. lim x

4x4 + 2

32 + 5 = _________

24. lim x

4x4 + 2

32 + 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. limx0

sin 6

3= ________________

2. limx0

7

cos(7)= ________________

3. limx0

tan

sin = _______________

4. lim x0

sin 5

sin 7= ________________

5. limx0

2 2cos()

= _____________

6. limx0

4 4 cos2

sin2 ________________

7. limx0

sin2 2

42= ________________

8. limx0

tan2(4 )

2= ________________

9. limx0

csc = _________________

10. lim x0

2

sin() 1= _____________

11. limx0

sin2

6= ________________

12. limx0

tan = _________________

13. limx0

1 sec

= _______________

14. limx0

4

sin = ________________

15. limx0

+ sin

sin = _______________

• 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 = 2 5 at = 0.

2. Calculate the slope of the tangent line

to = 22 4 + 4 at = 1.

3. Calculate the slope of the tangent line

to = 32 4 + 5 at = 1.

4. Calculate the slope of the tangent line

to = 3 at any value x.

5. Calculate the slope of the tangent line

to = 23 + 1 at = 2.

6. Calculate the slope of the tangent line

to =1

at = 1.

7. Calculate the slope of the tangent line

to = 2

at = 1.

8. Calculate the slope of the tangent line

to = 10 at any value x.

• NJCTL.org

9. Calculate the slope of the tangent line

to =1

+4 at = 2.

10. Calculate the slope of the tangent

line to =

2 at = 3.

11. Calculate the slope of the tangent

line to = at any value x.

12. Calculate the slope of the tangent

line to = + 3 at = 6.

13. Calculate the slope of the tangent

line to = at = 0.

14. ** Calculate the slope of the tangent

line to = at = 7.

• NJCTL.org

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

1. 2 2. -4

3. 8 = 4( 1

2) or =

4 + 6 4. 5 5. 1 6. = 7. 0 8. 2/3 9. 8 = 12( 3) or =12 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. 3 + 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

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

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