Worksheet 13
Limits and Their Properties Review
1
⎛ sin 5x ⎞
lim ⎜
⎟
x→0 ⎝ cos 4 x ⎠
2
⎛ sin 2 3x ⎞
lim ⎜ 2
x→0 ⎝ x cos x ⎟
⎠
3
⎛
⎞
⎜ sin 5x ⎟
lim ⎜
x→0
1 ⎟
⎜⎝ sin x ⎟⎠
3
1 + sin x
limπ
x→ 1 − cos x
4
⎛ x ⎞
lim ⎜
⎟
x→0 ⎝ tan x ⎠
6
cos 2x
x→∞
x2
7
⎛ 6x 3 − 5x ⎞
lim ⎜ 2
x→∞ ⎝ x − 3x ⎟
⎠
8
⎛ 4a 2 − x 2 ⎞
lim ⎜
x→b ⎝ 2a + x ⎟
⎠
9
8x 3 − 5x
x→−∞ x 2 − 3x
4x 3 − 32
lim 2
x→2 5x − 20
4 ⎛ x⎞
lim sin ⎜ ⎟
x→0 x
⎝ 5⎠
10
x2 + x4
x→∞ x 2 + x 6
5
lim−
x→4 x − 4
2x − 2
x→2 x − 4
Write an equation of the line perpendicular to
3x − 4y = 12 that passes through the point ( −3,1) .
16
5
2
11
13
15
17
lim
12
14
lim+
18
lim
lim
4x 3 − 5
x→1 5x 2 − 6
8x 2 − 2x 3
lim
x→∞ 2x 2 + 4x
lim
⎧ x 2 − 4, x ≤ −3
⎪
Graph f ( x ) = ⎨ x + 3, −3 < x < 2
⎪−x + 2, x ≥ 2
⎩
Use the following graph of f ( x ) to answer the next five questions.
19 lim+ f ( x )
x→1
20
lim− f ( x )
x→1
21
lim f ( x )
x→1
22
lim f ( x )
x→−1
23
lim f ( x )
x→2
Use the graph of f ( x ) below to answer 24 a-c
24 a) Use the 3-part definition of continuity to show if f ( x ) is continuous at x = 3
24 b) What type(s) of discontinuity are shown in the graph of f ( x ) ?
24 c) Is there is a removable discontinuity? If so, assign a value to remove it.
25
26
27
28
⎧2x − 1, x ≤ 1
If f ( x ) = ⎨
, use the definition of continuity to show if f ( x ) continuous at x = 1.
⎩−3x + 1, x > 1
⎧ x 2 + 6x + 8
, x ≠ −2
⎪
If f ( x ) = ⎨ x + 2
, use the definition of continuity to show if f ( x ) continuous at x = −2 .
⎪2, x = −2
⎩
x3 + 8
, use the definition of continuity to show if f ( x ) is continuous at x = −2 .
x+2
x 3 − 64
When f ( x ) =
, then f ( x ) has point discontinuity. Assign a value to f ( x ) that removes the discontinuity.
x−4
NOTATION!!
If f ( x ) =
Answers:
1. 0
2. 9
3. 15
8. 2a − b
9. −∞
10. 0
15. -1
16.
22. 2
23. 1
∞
17. y − 1 = −
4. 1
5. 2
12
5
18. Graph
below
11.
4
( x + 3)
3
18.
6. 0
12. −∞
19. 1
3
2
14. 1
7.
4
5
20. -2
13.
21. DNE
24.a)
I. f ( 3) = 3
II. lim f ( x ) exists ⎡⎢ lim− f ( x ) = lim+ f ( x ) = 2 ⎤⎥
x→3
⎣ x→3
⎦
x→3
III. f ( 3) ≠ lim f ( x )
x→3
∴ f ( x ) is not continuous at x = 3
24. b) removable at x = 3 , non-removable at x == 1
24. c) f ( 3) = 2
25.
I. f (1) = 1
26.
I. f ( −2 ) exists ⎡⎣ f ( −2 ) = 2 ⎤⎦
II. lim f ( x ) DNE ⎡⎢ lim+ f ( x ) = 1 ≠ lim_ f ( x ) = −2 ⎤⎥
x→1
⎣ x→1
⎦
x→1
∴ f ( x ) is not continuous at x = 1
II. lim f ( x ) exists ⎡⎢ lim+ f ( x ) = lim− f ( x ) = 2 ⎤⎥
x→−2
⎣ x→2
⎦
x→2
III. f ( −2 ) = lim f ( x ) = 2
27.
I. f ( −2 ) DNE
∴ f ( x ) is continuous at x = −2
28.
( x − 4 ) x 2 + 4x + 16
x 3 − 64
lim
= lim
= lim x 2 + 4x + 16 = 48
x→4 x − 4
x→4
x→4
(x − 4)
∴ f ( x ) is not continuous at x = −2
x→−2
(
f ( 4 ) = 48
)
(
)