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.