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AP CALCULUS
1001 - Limits 1: Local Behavior
You have 5 minutes to read a paragraph out of the provided magazine and write a thesis statement regarding what you read
C CONVERSATION: Voice level 0. No talking!
HHELP: Raise your hand and wait to be called on.
AACTIVITY: Whole class instruction; students in seats.
M MOVEMENT: Remain in seat during instruction.
PPARTICIPATION: Look at teacher or materials being discussed. Raise hand to contribute; respond to questions, write or perform other actions as directed.NO SLEEPING OR PUTTING HEAD DOWN, TEXTING, DOING OTHER WORK.
S
Activity: Teacher-Directed Instruction
Objectives(SWBAT): Content: evaluate limits using basic limit laws, direct substitution, factoring, and rationalizing Language: SW verbally describe limit laws in their own words
REVIEW:
ALGEBRA is a _________________ machine that ___________________ a function ___________ a point.
CALCULUS is a ________________________ machine that ___________________________ a function ___________ a point
function
evaluates
Limit
Describes the behavior of
near
Limits Review:
PART 1: LOCAL BEHAVIOR (1). General Idea: Behavior of a function very near the point where
(2). Layman’s Description of Limit (Local Behavior)
(3). Notation
(4). Mantra
x ax a
xa yL
L
a
G N A W Graphically
2x
Lim f x
1x
Lim f x
“We Don’t Care” Postulate”:
The existence or non-existence of f(a) has no bearing on the
What is the y value?
What is the y value?
0
3
G N A WNumerically
2
5 25 if ( )
2
x
xLim f x f x
x
21.9991.9 1.99x
y
2.0001 2.001 2.01
40.268 40.56140.20439.91437.165 40.239error
lim𝑥→2
5𝑥−25𝑥−2
≈ 40.2
C CONVERSATION: Voice level 0. No talking!
HHELP: Raise your hand and wait to be called on.
AACTIVITY: Whole class instruction; students in seats.
M MOVEMENT: Remain in seat during instruction.
PPARTICIPATION: Look at teacher or materials being discussed. Raise hand to contribute; respond to questions, write or perform other actions as directed.NO SLEEPING OR PUTTING HEAD DOWN, TEXTING, DOING OTHER WORK.
S
Activity: Teacher-Directed Instruction
Objectives(SWBAT): Content: evaluate limits using basic limit laws, direct substitution, factoring, and rationalizing Language: SW verbally describe limit laws in their own words
G – GraphicallyN – NumericallyA – AnalyticallyW -- Words
The Formal Definition
Layman’s definition of a limitAs x approaches a from both sides (but x≠a) If f(x) approaches a single # L then L is the limit
The function has a limit as x approaches a if, given any positive number ε, there is a positive number δ such that for all x, 0< < δ ε
FINDING LIMITS
G N A W
0
sin( )x
xLim
x
cos( ) 10
x o
xLim
x
-.1 -.01 -.001 0 .001 .01 .1
X
0
0
Mantra:
• Numerically
• Words
Verify these also:
0
11
x
x
eLim
x
xa, yL
.9999 .99999 .9834.99834 .999 .99999
Must write every time
(6). FINDING LIMITS
“We Don’t Care” Postulate…..• The existence or non-existence of f(x) at x = 2 has
no bearing on the limit as x a
2( ) 2 1f x x x 3 22 2 4
( )2
x x xf x
x
• Graphically 2x
FINDING LIMITS
• Analytically
A. “a” in the Domain
Use _______________________________ 3
3
1
1x
xLim
x
B. “a” not in the Domain
This produces ______ called the _____________________ 3
1
1
1x
xLim
x
Rem: Always start with Direct Substitution
Direct substitution
13
00
Indeterminate form
00
Rem: Always start with Direct Substitution
Method 1: Algebraic - Factorization
4
2 0
4 0x
xLim
x
Method 2: Algebraic - Rationalization
3
1
1 0
1 0x
xLim
x
Method 3: Numeric – Chart (last resort!)
3
0
1 0
0
x
x
eLim
x
Method 4: Calculus
To be Learned Later !
Creates a hole so you either factor or rationalize
{¿
Do All Functions have Limits?Where LIMITS fail to exist.
0
1, 0
3, 0x
xLim
x
2
4
2x
xLim
x
0
1sin
xLim
x
Why?
0xLim x
f(x) approaches two different numbers
Approaches ∞ Oscillates
At an endpoint not coming from both sides
Review :1) Write the Layman’s description of a Limit.
2) Write the formal definition. ( equation part)
3) Find each limit.
4) Does f(x) reach L at either point in #3?
4( )
xLim f x
4( )
xLim f x
Using Direct Substitution
BASIC (k is a constant. x is a variable)
1)
2)
3)
4)
x aLimk k
x aLim x a
n n
x aLim x a
( ) ( )x a x aLim kx k Lim x
IMPORTANT: Goes
BOTH ways!
Properties of Limits
Properties of Limits: cont.
POLYNOMIAL, RADICAL, and RATIONAL FUNCTIONS
all us Direct Substitution as long as a is in the domain
OPERATIONSTake the limits of each part and then perform the operations.
EX: 2 2
3 3 3(2 4 ) 2 4
x x xLim x x Lim x Lim x
Composite Functions
REM: Notation
THEOREM:
and Use Direct Substitution.
( )f g x f g x
( ( )) ( ( ))x a x aLim f g x f Lim g x
EX: EX:
2
1xLim x
x
sin( )
6
x
x
Lim e
Limits of TRIG Functions
Squeeze Theorem: if f(x) ≤ g(x) ≤ h(x) for x in the interval about a, except possibly at a and the
Then exists and also equals L
( ) ( )x a x aLim f x L Limh x
( )x aLim g x
f
g
h
a
This theorem allow us to use DIRECT SUBSTIUTION with Trig Functions.