CALCULUS AB
CALCULUS AB
WORKSHEET 1 ON LIMITS
Work the following on notebook paper. No calculator.
1. The graphs of f and g are given. Use them to evaluate each limit, if it exists. If the limit does not exist, explain why.
[pic]
____________________________________________________________________________
Find the following limits. Show all steps.
2. [pic]
3. [pic]
4. [pic]
5. [pic]
6. [pic]
7. [pic]
TURN--->>>
8. Graph [pic] on the same graph over the x-interval from
[pic] to 1, and use the Squeeze Theorem to find [pic].
9. Sketch the graphs of [pic], where f is any continuous
function that satisfies the inequality [pic] for all x in the interval
[pic]. What can you say about the limit of [pic] Explain your
reasoning.
10. If [pic]
____________________________________________________________________________
Evaluate. Show all steps.
11. [pic]
12. [pic]
13. [pic]
14. [pic]
Answers to Worksheet 1 on Limits
1. (a) 2 (b) dne (c) 0 (d) undefined
(e) 16 (f) 2
2. 2
3. 1
4. 2
5. 0
6. 4
7. 2
8. 0
9. [pic] by the Squeeze Theorem.
10. [pic] by the Squeeze Theorem.
11. 4
12. [pic]
13. [pic]
14. 48
CALCULUS AB
WORKSHEET 2 ON LIMITS
Find the limit. Draw a sketch for each problem. Do not use your calculator.
1. [pic] 2. [pic]
3. [pic] 4. [pic]
5. [pic] 6. [pic]
7. [pic] 8. [pic]
9. [pic] 10. [pic]
[pic]
12. [pic]
[pic]
[pic]
15. [pic] 16. [pic]
17. [pic] 18. [pic]
TURN--->>>
On problems 19 - 24:
(a) find [pic]
(b) find [pic]
(c) identify all horizontal asymptotes.
Use your graphing calculator on problems 23 and 24.
19. [pic]
20. [pic]
21. [pic]
22. [pic]
Hint on 22: Use the definition of absolute value, [pic]
23. [pic]
24. [pic]
Answers to Worksheet 2 on Limits
1. [pic]
2. dne
3. [pic]
4. [pic]
5. [pic]
6. [pic]
7. dne
8. dne
9. 8
10. [pic]
11. (a) 3
(b) 4
(c) dne
12. [pic]
13. [pic]2
14. 4
15. [pic]
16. [pic]
17. [pic]
18. [pic]
19. (a) [pic]
(b) [pic]
(c) no horizontal asymptotes
20. (a) 0
(b) 0
(c) y = 0
21. (a) 3
(b) 3
(c) y = 3
22. (a) 3
(b) – 3
(c) y = 3 and y = – 3
23. (a) 0
(b) 0
(c) y = 0
24. (a) 1
(b) 1
(c) y = 1
CALCULUS AB
WORKSHEET 3 ON LIMITS
Evaluate the following. Show supporting work for each problem.
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
7. [pic]
8. [pic]
[pic]
(a)[pic]
(b)[pic]
(c)[pic]
TURN->>>
[pic]
[pic]
11. [pic]
13. [pic]
14. [pic]
15. [pic]
16. [pic]
17. [pic]
______________________________________________________________________________
18. If [pic] for all x, find [pic]. Which theorem did you use?
Answers to Worksheet 3 on Limits
1. [pic] 2. 8 3. [pic] 4. [pic] 5. [pic]
7. [pic] 8. – 1 9. (a) 0 (b) 1(c) dne 10. 3 11. – 9
13. 1 14. [pic] 15. [pic] 16. [pic] 17. 0
18. 2 by the Squeeze Theorem
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