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. ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )

2 1

0 1

2 1

3

a lim b lim

c lim d lim

e lim f lim 3

x x

x x

x x

f x g x f x g x

f xf x g x

g x

x f x f x

→ →

→ →−

→ →

⎡ ⎤ ⎡ ⎤+ +⎣ ⎦ ⎣ ⎦

⎡ ⎤⎣ ⎦

+

____________________________________________________________________________ Find the following limits. Show all steps.

2. ( )0

sin 2limx

xx→

3. 0 2sinlim2x

xx x→ −

4. 0

sinlimx

x xx→

+

5. 2

0

sinlimx

xx→

6. ( )( )0

3sin 4lim

sin 3x

xx→

7. 0

2lim1 cosx

xx→ −

TURN--->>>

8. Graph 50, , and cosy x y x y xxπ⎛ ⎞= = − = ⎜ ⎟⎝ ⎠

on the same graph over the x-interval from

1− to 1, and use the Squeeze Theorem to find 0

50lim cosxx

xπ

→

⎛ ⎞⎜ ⎟⎝ ⎠

.

9. Sketch the graphs of ( )21 , cos , and y x y x y f x= − = = , where f is any continuous

function that satisfies the inequality ( )21 cosx f x x− ≤ ≤ for all x in the interval

,2 2π π

−⎛ ⎞⎜ ⎟⎝ ⎠

. What can you say about the limit of ( ) as 0?f x x→ Explain your

reasoning. 10. If ( ) ( )

1

33 2, evaluate lim .x

x f x x f x→

≤ ≤ +

____________________________________________________________________________ Evaluate. Show all steps.

11. 2

23

2 154 3

limx

x xx x→−

− −+ +

12. 7

2 37

limx

xx→

+ −−

13. 0

1 15 5lim

xxx→

−+

14. 3

4

644

limx

xx→

−−

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. ( )

0lim 1x

f x→

= by the Squeeze Theorem.

10. ( )1

lim 3x

f x→

= by the Squeeze Theorem.

11. 4

12. 16

13. 125

−

14. 48

CALCULUS AB WORKSHEET 2 ON LIMITS Find the limit. Draw a sketch for each problem. Do not use your calculator.

1. 1

11

limx x+→

=−

2. 111

limx x→

=−

3. ( )2313

limx x→−

=+

4. 5

15

limx x−→

=−

5. ( )25

15

limx x−→

=−

6. ( )22

12

limx x→

− =−

7. 3

33

limx

xx→

−=

− 8. ß ®2

1limx

x→

+ =

9. 3

2

22

limx

x xx+→

−=

− 10. ß ®3

4

44

limx

x xx−→

−=

−

( )

( )( )( )

2

2

2

2

1 if 211.

3 2 if 2

)

)

)

lim

lim

lim

x

x

x

x xf x

x x

a f x

b f x

c f x

−

+

→

→

→

⎧ − <= ⎨

− >⎩=

=

=

12. 3

133

limx

xx+→

⎛ ⎞− − =⎜ ⎟−⎝ ⎠

( ) ( )1

3 if 113. lim

4 if 1 x

x xg x g x

x →

− ≠⎧= =⎨ =⎩

( ) ( )2 1

3 if 114.

3lim

1 if 1 x

x xh x h x

x x →

+ <⎧= =⎨⎩ + >

15.

2

tanlimx

xπ +

→= 16.

2

seclimx

xπ +

→−=

17. csclim

xx

π −→= 18.

0cotlim

xx

−→=

TURN--->>>

On problems 19 - 24: (a) find ( )lim

xf x

→∞

(b) find ( )limx

f x→−∞

(c) identify all horizontal asymptotes. Use your graphing calculator on problems 23 and 24.

19. ( )33 13

x xf xx− +=+

20. ( )2

34 3 52 1x xf xx x− +=+ −

21. ( ) 3 14

xf xx+=−

22. ( ) 3 12

xf xx+=+

Hint on 22: Use the definition of absolute value, if 0if 0

x xx

x x≥⎧

= ⎨− <⎩

23. ( ) ( )sin 3xf x

x=

24. ( ) 1cosf xx

⎛ ⎞= ⎜ ⎟⎝ ⎠

Answers to Worksheet 2 on Limits 1. ∞ 2. dne 3. ∞ 4. ∞ 5. ∞ 6. −∞ 7. dne 8. dne 9. 8 10. ∞ 11. (a) 3 (b) 4 (c) dne 12. −∞ 13. − 2 14. 4 15. −∞ 16. ∞ 17. ∞ 18. −∞ 19. (a) ∞ (b) ∞ (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. 2

24

43 4

limx

x xx x→

− =− −

2. ( )20

4 16limx

xx→

+ −=

3. 0

9 3limx

xx→

+ − =

4. 0

1 12 2lim

xxx→

−+ =

5. 2

32

48

limx

xx→

− =−

6. ß ®

21lim

xx

+→+ =

7. 3

13

limx x−→

=−

8. 4

44

limx

xx+→

− =−

( ) 2

1 if 19.

if 1x x

f xx x− ≤⎧

= ⎨>⎩

(a) ( )

1limx

f x−→

=

(b) ( )

1limx

f x+→

=

(c) ( )

1limx

f x→

=

TURN->>>

( ) 2 if 110.

if 1x x

g xxπ

+ ≠⎧= ⎨ =⎩

( )

1limx

g x→

=

11. 2

3

33

limx

x xx−→

−=

−

12. ß ®2

3

33

limx

x xx−→

−=

−

13. 0

tanlimx

xx→

=

14. 20

sin7 3

limx

xx x→

=−

15. 0

4 sin3

limx

x xx→

+ =

16. ( )( )0

2sin 5sin 4

limx

xx→

=

17. ( )20

1 cos5

limx

xx→

−=

______________________________________________________________________________ 18. If ( ) 242 2x g x x x≤ ≤ − + for all x, find ( )

1limxg x

→. Which theorem did you use?

Answers to Worksheet 3 on Limits

1. 45

2. 8

3. 16

4. 14

−

5. 13

6. 3 7. ∞ 8. – 1 9. (a) 0 (b) 1 (c) dne 10. 3 11. – 9 12. ∞ 13. 1

14. 17

15. 53

16. 52

17. 0 18. 2 by the Squeeze Theorem