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AP CALCULUS AB
LIMIT WORKSHEET #1
Find the indicated limit. Which method is most appropriate: Direct Substitution, Numerical,
Analytic or Graphical?
lim (3x  2)
1.
x  3
x 0
2.
x3  1
x  1 x  1
3.
2 x2  x  3
x  1
x 1
4.
lim
x 1
x
lim
x
x  2x  3
lim
lim
5.
x 0
x 3 
7. lim
2
( x  x)3  x3
6. lim
x  0
x
sin 4 x
sin 2 x
1
1
x
8. lim
x 1 x  1
 2  2
9. lim
  0 sin 2
s 1
 s
10. lim f (s) ; where f ( s)  
s1
1  s s  1
s3
 s
11. lim
f (s) ; where f ( s)  
3
s1
6  s s  3
Find the discontinuities (if any) for each function. Identify which are Removable and which
are Nonremovable Jump or Nonremovable Infinite? Analyze each initially without a graph,
then draw a sketch afterwards to confirm.
12. f ( x) 
1
x 1
2 x  3 x  1
17. f ( x)  
2
x 1
 x
13. f ( x) 
x
x 1
 x5

18. f ( x)   3
3  x 3
2
2
14. f ( x)  [[ x  3]]
15. f ( x) 
3x 2  x  2
x 1
16. f ( x) 
x 1
2x  2
ANSWERS
1.
2.
3.
4.
5.
6.
7.
8.
-7
1
-5
–

3x^2
2
-1/2
cos x
19. f ( x)  
x  2
x 1
x 1
x0
x0
9. 1
10. DNE
11. 3
12. None, contin (-,)
13. Nonremovable @ x = +/-1
14. Nonremovable @ every integer
15. Removable @ x = 1
16. Removable @ x = -1
AP CALCULUS AB
LIMIT WORKSHEET #2
Find the indicated limit.
1. lim
0
2. lim

tan 2 y
3y
sin 
9. lim
tan 
10. lim cot 2

0
y 0
0
sin 2 x
2x2  x
sin x
x 0 3x
11. lim
sin 5 x
x  0 sin 3 x
12. lim tan 2 x csc 4 x
3. lim
4. lim
x 0
x 0
13. limcot x sin 4 x
x
5. lim
x  0 sin 3 x
6. lim
sin 2 

0
7. lim
1  cos 

0
x 0
1  cos 2 x
x 0
x2
14. lim
tan 3x
x  0 tan 2 x
15. lim
x2
x  0 sin x
16. lim
3x
x  0 sin 5 x
8. lim
Answers:
1.
1
2.
1
3.
1
4.
5
5.
1
6.
0
14. 1
7.
0
15. 3
8.
3
3
9.
2
10.
1
2
11. 2
3
12.
3
13. 4
5
3
1
16. 0
2
2
AP CALCULUS AB
LIMIT WORKSHEET #3
Find the indicated limit.
x 2  3x
x 0
x
3 3cos x
1. lim 
x 0 x
x
11. lim
2. lim tan x
12. lim  sec
x 
3. lim sec
x 7
 
x
13. lim
6
x 4
 x
4. lim f ( x); where f ( x)  
x 1
2  x
(Graphically)
x 1
x 1
;
tan 2 x
14. lim
x 0
x
15. lim(1  cos 2h)
h 0
2 x
5. lim 2
x 2 x  4
5 x3  1
6. lim
x   10 x 3  3 x 2  7
16. lim
x  1
x 1  2
7. lim
x 3
x3
lim
x  3
2
x2
9. lim sin
x 1
x 1
2
5( x  x)2  5 x 2
x  0
x
18. lim
x3  2 x 2  3x  1
x 
x2  x  2
19. lim
x
20. lim( x  3) 2
2
 1
10. lim 1  
x 
 x
x 1
17. lim  sec x
x 
8.
x 2
x4
x 
x
Find the interval for which the function is continuous.
21. f ( x)  x x  3
22. f ( x) 
x 1
x
23. f ( x) 
24. f ( x) 
1
( x  1) 2
1
x 4 3
Find the discontinuities (if any) for the given function. State whether they are Removable,
Nonremovable Jump or Nonremovable Infinite. Use your TI-83/84 to verify your responses.
25. f ( x) 
26. f ( x) 
1
x 4
2
x 1
x  x2
2
27. f ( x)  cos
x
2
x 1
 x

28. f ( x)   2
x 1
2 x  1 x  1

 x  3
29. f ( x)   2
 x  x  6
x 3
x 3
x2
 2 x
30. f ( x)   2
x  4x 1 x  2
 x
 tan
31. f ( x)  
4
 x

x 1
x 1
Answers:
1.
1
16. 1
2.
0
17. 
3.
2
4.
1
5.
1
6.
1
7.
1
8.
-2
9.
1
18. 10x
3
20. 0
4
4
11. -3
12. 
13.
14. 0
15. 2
21. [3, )
22. (0, )
2
10. 2.718 … “e”
1
19. 
4
23. (,1)  (1, )
24. (4,13)  (13, )
25. Continuous for (, )
26. 1, Removable; -2 Nonremovable Infinite
27. Continuous for (, )
28. 1, Removable
29. Continuous for (, )
30. 2, Nonremovable Jump
31. Continuous for (, )
AP CALCULUS AB
LIMIT WORKSHEET #4
Find each of the following limits.
 13
x 3 x  3
4 x3  6 x 2  3
x  5 x3  7 x 2  9
13. lim
2. lim
9 x4  7 x2  8x
x   4 x 5  3 x  12
14. lim
3x3  7 x 2  5 x  1
3. lim
x 
7 x2  2 x  5
15. lim [[ x]]
1. lim
(1) x
4. lim
x 
x
5. lim sec
x 
1
x
x 2  25
x 5 x  5
1
x
2  2 x
x
x 0
x 3
16. lim[[ x  2]]
x 3
1
x 3 x  3
17. lim
18. lim
x 3
1
x3
6. lim
x4
7. lim
x 4
x 2
x2  5x  6
x 3
x 3
8. lim
x3  1
9. lim
x  1 x  1
x
10. lim
x 0 x
11. lim
x 4
x4
x4
2 x
x  2 x
12. lim
19. lim tan x
x 
2
20. lim csc x
x 
1
21. lim  
x  3
 
x
22. lim f ( x) ; where
x 1
 2x x  1
f ( x)  
3  x x  1
 x2 x  2
23. lim g ( x) ; where g ( x)  
x 2
2 x x  2
Answers:
1.
4
2.
0
3.

5
13.  1
14.
9
1
2 2
15. 3
4.
0
5.
1
6.
10
7.
4
8.
1
9.
3
16. 4
17. Does not exist
18. 
19. Does not exist
20. 
21. 0
10. 1
22. 2
11. -1
23. Does not exist
12. 0