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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
7/24/2019 AP Calculus Study Sheet 03
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7/24/2019 AP Calculus Study Sheet 03
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f x fx
x y x2 y cosx x 2 cosx 0
x .82413, .82413
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