Limits and Continuity Worksheet

LIMITS WORKSHEET #1

Find the indicated limit. Which method is most appropriate: Direct Substitution,Numerical, Analytic or Graphical?

1.3

lim (3 2)x

x

2.3

1

1lim1x

xx

3.2

1

2 3lim1x

x xx

4.0

1limx

xx

5. 23lim 2 3xx

x x

6.3 3

0

( )limx

x x xx

7.0

sin 4limsin 2xxx

8.1

1 1lim

1xxx

9.2

0

2limsin 2

10.1

lim ( )sf s

; where

1( )

1 1s s

f ss s

11. 1lim ( )sf s

; where3

( )6 3s s

f ss s

Find the discontinuities (if any) for each function. Identify which are Removable andwhich are Nonremovable Jump or Nonremovable Infinite? Analyze each initiallywithout a graph, then draw a sketch afterwards to confirm.

12. 21( )

1f x

x

13. 2( ) 1xf xx

14. ( ) [[ 3]]f x x

15.23 2( )

1x xf xx

16. 1( )2 2xf xx

17. 22 3 1

( )1

x xf x

x x

18.3

5 1( ) 3

3 1

x xf x

x x

19.cos 0

( )2 0x x

f xx x

ANSWERS:

1. -72. 13. -54. 5. 6. 3x^27. 28. -1/29. 110. DNE11. 312. None, contin (-,)13. Nonremovable @ x = +/-1

14. Nonremovable @ every integer15. Removable @ x = 116. Removable @ x = -1

3

LIMITS WORKSHEET #2 Find the indicated limit.

1.0

limsin

2.0

tanlim

3.0

sinlim3xxx

4.0

sin 5limsin 3xxx

5.0

limsin 3xxx

6.2

0

sinlim

7.0

1 coslim

8.0

3limsin 5xxx

9.0

tan 2lim3yyy

10.0

lim cot 2

11. 20sin 2lim2x

xx x

12.0

lim tan 2 csc 4x

x x

13.0

lim cot sin 4x

x x

14.2

20

1 coslimx

xx

15.0

tan 3limtan 2xxx

16.2

0lim

sinxxx

Answers:

1. 1

2. 1

3. 13

4. 53

5. 136. 0

7. 0

8. 35

9. 23

10. 1211. 2

12. 1213. 4

14. 1

15. 3216. 0

LIMITS WORKSHEET #3 Find the indicated limit.

1.0

3 3coslimx

xx x

2. lim tanx

x

3.7

limsec6xx

4.1

lim ( );xf x

where

1( )

2 1x x

f xx x

;

(Graphically)

5. 222lim

4xx

x

6.3

3 2

5 1lim10 3 7x

xx x

7.3

1 2lim3x

xx

8.3

2lim2x x

9.1

limsin2xx

10. 1lim 1x

x x

11.2

0

3limx

x xx

12. lim sec

13.4

2lim4x

xx

14.2

0

tanlimx

xx

15.0

lim(1 cos 2 )h

h

16.1

1lim

1xxx

17.2

lim secx

x

18.2 2

0

5( ) 5limx

x x xx

19.3 2

2

2 3 1lim2x

x x xx x

20. 2lim( 3)xx

Find the interval for which the function is continuous.

21. ( ) 3f x x x

22. 1( ) xf xx

23. 21( )

( 1)f x

x

24. 1( )4 3

f xx

Find the discontinuities (if any) for the given function. State whether they areRemovable, Nonremovable Jump or Nonremovable Infinite. Use your TI-83/84 to verifyyour responses.

25.2

1( )4

f xx

26. 21( )

2xf x

x x

27. ( ) cos2xf x

28.1

( ) 2 12 1 1

x xf x x

x x

29. 23 3

( )6 3

x xf x

x x x

30. 22 2

( )4 1 2x x

f xx x x

31.tan 1

( ) 41

x xf x

x x

Answers:

1. 1

2. 0

3. 23

4. 1

5. 14

6. 12

7. 148. -2

9. 1

10. 2.718 e

11. -3

12.

13. 1414. 0

15. 2

16. 1

17. 18. 10x

19. 20. 0

21. [ 3, )

22. (0, )

23. ( ,1) (1, )

24. (4,13) (13, )

25. Continuous for ( , )

26. 1, Removable; -2 Nonremovable Infinite


28. 1, Removable


30. 2, Nonremovable Jump


LIMIT WORKSHEET #4 Find each of the following limits.

1.3 2

3 2

4 6 3lim5 7 9xx xx x

2.4 2

5

9 7 8lim4 3 12xx x xx x

3.3 2

2

3 7 5 1lim7 2 5xx x xx x

4. ( 1)limx

x x

5. 1lim secx x

6.2

5

25lim5x

xx

7.4

4lim2x

xx

8.2

3

5 6lim3x

x xx

9.3

1

1lim1x

xx

10.0

limx

xx

11.4

4lim

4xxx

12. 2lim2

x

xx

13.1 1

3

3lim

3x

x x

14.0

2 2limx

xx

15.3

lim [[ ]]x

x

16.3

lim [[ 2]]x

x

17.3

1lim3x x

18.3

1lim3x x

19.2

lim tanx

x

20. lim cscx

x

21. 1lim3

x

x

22.1

lim ( )xf x

; where

2 1( )

3 1x x

f xx x

23.2

lim ( )xg x

; where

2 2( )

2 2x x

g xx x

Answers:

1. 452. 03. 4. 05. 16. 107. 48. 19. 310. 111. -112. 0

13. 19

14. 12 2

15. 316. 417. Does not exist18. 19. Does not exist20. 21. 022. 2 23. Does not exist

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Limits and Continuity Worksheet

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