AP Calculus AB Study Guide

Unit : L nit 4 :

im its + Unit 2 : L nit 3: Differential

Continuity Differentiation Intergrat Equations

1 M 1




2EC) V99L JE :

ABC ABC Procedures

& + -



11) Equations/theorems
ABC Terms




Example

, v :


v
T -




Say : "Limit of f(x) ,
& methods of evaluate Limits :

as X approach c is L + Table (Ex 1)

Graph (Ex2)
Notation: Limf(x)L
+


+ Algebraic (Ex 3 ,
4 ,
5)
facior' conjugates tris identities

Table :


Ex : WhatDoes y approach as
X get closer 10-1 ?


x + 5x + 6 = + 3
a) f(x) = b) f(x) =




↑ #
x + 2




&




X f(x) X f(x)


-

2 .
1 0 .
9 Y
-

2 .
1 -
9 Y

-

1 .
01 0 99 .
~
-

1 .
01 -
99 ~




-
2 001
.
0 .
999 ~ 1 -
2 . 001 -
999 -
-

1000

-
L N/A -
L N/A M
M

-
1 .
995 1 .
001 1 -
1 .
995 1001 10001
1 1

-
1 99 .
1 .
01 -
1 99.
101

-
1 5 1
. 1 Y -
1 5 11 Y
. .




x 5x 6 Lim X + 3
Lim + +

= 1 = DNA
2 Xc 2 2
2
-



Xc -


+ X +
x

,graphi




f(x) Lim f(x) = L




- -
X
-
- a

>
-


a I
⑱ f(a) = M

-
-




I
Y = a



Lim f(x) = DNE
x -
a



-
-

Ex 2 :




- -




I I .




I
I




I ↑
Lim f(x) = 1 Lim f(x) = DNE Lim f(x) I 3
X =
3 X - 6 X S 0-
Lim f(x) = DNE
X ,
0


Lim f(x) = 1 5 . Live f(x) = 3 Lim f(x) =
- 3
X >
Ot
x
-
4 X(




Algebraic :
Strategy : w = = b = real# factor

Lim -(t) c f (a) Coningate
Xia
Direct sub = 0/0= Intermedian Trig Identities

form

, factor :



Excim
~
Ex x If

99.9
: Lim -


1

I
X13
X -
3 S



Lim 1 Lim X 1
3)(x 3)
x



X
-




(x
-




Lim Lim +
-



I
x -
1 x -
1
X13
=
X13 x3 -
1 (x -
1)(x2 + x + 1)
X -
3




·
Lim
Lim x + 3 I
=
X13


= 3 + 3 = 6




&
I 1




Conjugate :

Ex :
~

Lim x + 1 -
11 -
1 8 If
- =
X = 8
X 8 G



Lim X + 1 1 Lim X + 1 1
( in In
- -




-
X = 8 X = 8
X X




( x + 11)(x + 1 + 1) Lim X Lim 1
= =
0
= X x + 1 +
x + 1 + 1 -

1
x ,

x(x + 1 1)
+
X > O
(x + 1 + 1)


1 1 1
I = I

(0 + 1 + 1) (x + 1) 2


Tris Identities :



Ex : O

1 cos
1 O If

==
Lim -

cos
-
X =
0 T
2 Sir O O




Cos
Lim 1 -

cos O Lim 1 -
cos &
X =
0 = X ·
o
2 sin
"
O (1 + cos@)(1-cosO)

-cost 1 + =




sin 1 -cos"E
~
= Lim 1
I
X O
(1 + cos@)



1 1 1
I = I

1 + cosO 1 + 1 2

, I
Limit Properties :

Lim

x - C
f(x) = g(x) =


x
Lim
- C
f(x) ! Lim

x - C
g(x)
Assumeinc a
= 1 + M

Lim f (x) · g(x) = Lim f(x) · Lim g(x) Limkf(x) = k Lim f(x)
x - C x - C x - C x - C x - C



I L .
M = kL
T
Lim(f(x))
(Lim f(x)
Lim
cf(x) =




imI
X ·
L x - C x - C
= -
M
Lim M
g(x) I L
S


X > C




Composite func
: im f(g(x) =
f(ximal
>
If : im g(x) = M exist + continue at M




Ex : Lim
s(h(x)) = ?
X 3 -
1

1 1

Lim
O 3 3
Lim
· (n(x)) = 9(h(x)) = 3
1-
-




2 2
x3 -

n(x)c j
9 h

1 1

Lim
Lim
9(h(x))
L
-
3 - 2 -
l
-
1
1 2
·
3
& 2
-
3 - 2 -
l
-
1
1 2 3
&
x . -1
I
- =
n(x)c - 2S(n(x)) = 3


⑭ 2 8 O 2
Lim
g(n(x))
- -




= 3
-
S ⑧ -
S X 3 -
1
~ ~




Limits x Piecwise function :



I
+ + 4x))
Ex : Let +(x) = firD
Lim
f(x)
12. x 2
3x 1
.
+ x




↑ (x

↓
Lim
. 2
-
3x + 1 = 7

~
im f(x) = DNE




/
Lim
2
+ x + 4 = S
x .




-




2