AP Calculus Study Sheet 03

7/24/2019 AP Calculus Study Sheet 03

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limx3

gx

gx x29x3

gx |x 3 |

gx |x3|

x3


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-20

-15

-10

-5

0

5

1 2 3 4 5x

x 2.9 2.99 2.999 3.001 3.01 3.1

gx 4.41 4.9401 4.994 5.006 5.0601 5.61

limx3

gx limx3

gx limx3

gx.

limx3

3

limx3

3

limx3

gx 5.

limx3

gx 5.

limx3 gxDNE.


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y gx

-1

0

1

2

3

4

-3 -2 -1 1 2 3x

limx1

gx

limx 3

2

gx

limx1

gx

limx1

gx

limx1

gx

limx2

gx

limx1

gx 3.

limx 3

2

gx 1.

limx1

gx DNE.

limx1

gx 3.

limx1

gx 0.

limx2

gx 3.

limx1

x1

x21 lim

t0

t2 9 3

t2


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t limt0

t2 9 3

t2

0 t

limx1

x1

x21

12

limt0

t2 9 3

t2

16

limxa

fx fa

limv1

v2 4v 2

limx5

x2 25

limx1.5

x 2

lim

v1v2 4v 2 7

limx5

x2 25 0

limx1.5

x 2 1

limh0

4 16h

h

limx2

x3 8

x 416

limx2

1x

1

2

x2

limxb

x2b2

xb


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limh0

4 16h

h lim

h0

4 16h

h

4 16h

4 16h lim

h0

1616h

h 4 16h lim

h0

1

4 16h

18

limx2

x3 8

x416 lim

x2

x2 x 22x4

x2x2 x2 4 lim

x2

x22x4

x2 x2 4

38

limx2

1x

1

2

x2 lim

x2

2 x2x

x2 lim

x2

12x

14

limxb

x2b2

xb lim

xb

xbxbxb

limxb

x b 2b

limxa

k k.

limxa

x a.

limxa

fx hx limxa

fx limxa

hx.

limxa

fx hx limxa

fx limxa

hx.

limxa

fxhx

limx a

fx

limx a

hx lim

xahx 0.

limxa

|fx| limxa

fx.

limxa

kfx klimxa

fx.

limxa

fx fa fx

limxa

|fx| limxa

fx

limxa

|fx| limxa

fx limx2

|3x | 6 limx2

3x 6


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lim

6

tancos

limx0

1cosxsinx

limx0

xsin8x

limx02x1cosx

4x

limx0

1cosxx 0 limx0

sinxx 1

lim

6

tancos

2

3

limx0

1cosxsinx

limx0

x1cosxx sinx

limx0

1cosxx limx0

xsinx

0 1 0

lim

x0

x

sin8x

1

8 lim

x0

8x

sin8x

1

81 1

8

limx0

2x1cosx4x

limx0

2x4x

1cosx

4x lim

x0

2x4x

limx0

1cosx4x

limx0

12

1

4 limx0

1cosxx

12

1

40

12

fx

-5

-4

-3

-2

-1

0

1

2

3

4

5

-5 -4 -3 -2 -1 1 2 3 4 5 6 7 8

limx1

fx

limx1

fx

limx1

fx

limx1

fx

limx4

fx


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limx4

fx

limxa

fx

limx1

fx .

limx1

fx .

limx1

fx .

limx1

fx .

limx4

fx .

limx4

fx .

limx

sinxx

sinx 1 1 x sinxx

limx

sinxx 0

limx

3x2x4

xx2

limx2

x2

x24

limx2

x2

x24

limx2

x2

x24

limx

3x2x4

xx2 lim

x3x 2x4

xx2

1

x2

1

x2

limx

3 1x 4

x2

1x 1

3

limx2

x2

x24 lim

x2

x2x2x2

limx2

1

x2


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limx2

x2

x24 lim

x2

1x2

limx2

x2

x24 undefined

-10

-8

-6

-4

-2

0

2

4

6

8

10

-5 -4 -3 -2 -1 1 2 3 4 5 6 7 8

limx

fx a limxa

fx

limx

fx 2 limx

fx 0 limx2

fx limx2

fx limx0

fx limx0

fx

limx4

fx limx4

fx .

limx

fx 3 limx

fx 0 limx0

fx limx0

fx limx3

fx limx3

fx .


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-8

-6

-4

-2

0

2

4

6

8

-4 -2 2 4x

limx

x47x9

45x3x4

limx

2xx3

x 23x

limx0

x37x9

7x x3

limx

2xx3

x2 2x

limx

x3

x2 x

limx

x5 5x

x

limx

x47x9

45x3x4

13

limx

2xx3

x 23x 0

limx0

x37x9

7x x3 9 0 0

limx

2xx3

x2 2x 1

limx

x3

x2 x 0

limx

x5 5x


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ax x4 3x 9

bx 4x5

2

cx x9 3x

dx 100x

ex e

fx ex

gx 2x x 3

hx 9x2

dx 100x

fx ex

gx 2x x 3

cx x9 3x

ax x4 3x 9

bx 4x5

2

hx 9x2

ex e

ht 2t 1.1 t t3

ht 1.9 133.39 1017.4 1.2498 10 5 9.8642 105

t 1 5 10 50 100

limt

ht

x

limt

ht 1.1t


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4.5 40 5

Vt At

Ct

limt

Ct

Vt 40 5t

At 4.55t 22.5t 5 4.5

Ct At

Vt

22.5t405t

limt

Ct

limt

Ct limt

22.5t405t

limt

22.5t405t

1t1t

limt

22.540

t 5

22.5

5 4.5

40

4.5

Ct At

Vt

22.5t405t

0

1

2

3

4

5

10 20 30 40 50t


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fx xc2 c

x c 0

f

f x 0 f

c c c

x x : x c2,x 0

x 0 limxa

fx fa limx0

fx

limx0

xc 2 cx limx0

xc2 cx

xc 2 c

xc 2 c lim

x0

xc2c2

x xc 2 c lim

x0

x

x xc2 c lim

x0

1

xc2 c

12c

12c x 0

x 0

limx

6xn3x4

ax 34x2 9 3 a n

n

3 0

a 2

Nt 500 t22

250t t : 0 t 12 t

u


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f x fx

x y x2 y cosx x 2 cosx 0

x .82413, .82413


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AP Calculus Study Sheet 03

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