AP CALCULUS

1002 - Limits 1: Local Behavior

REVIEW:

ALGEBRA is a ________________________ machine that ___________________ a function ___________ a point.

CALCULUS is a ________________________ machine that ___________________________ a function ___________ a point

  

 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)

 L

a

(3). Notation

(4). Mantra

x ax a

G N A W Graphically

2x

Lim f x

1x

Lim f x

“We Don’t Care” Postulate”:

G N A WNumerically

2

5 25 if ( )2

x

xLim f x f x

x

The Formal Definition

 (5). Formal Definition ( Equation Part)  

Graphically:Find a

If 3 2 1

1 2 3 4

0 0.5

3( )

xLim f x

• Analytically 

Find a if

given and

for

------------------------------------------

0 0.002

2(2 5 ) 12

xLim x

( ) 2 5f x x 2a 12L

Find a for any 0 2

13

9 1 23 1x

xLimx

Day 2

FINDING LIMITS

G N A W

0

sin( )x

xLimx

cos( ) 1 0x o

xLimx

-.1 -.01 -.001 0 .001 .01 .1

X

00

Mantra:

• Numerically

• Words

Verify these also:

0

1 1x

x

eLimx

(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( )

2x x xf x

x

• Graphically 2x

FINDING LIMITS

• Analytically

A. “a” in the Domain

Use _______________________________ 3

3

11x

xLimx

B. “a” not in the Domain

This produces ______ called the _____________________ 3

1

11x

xLimx

Rem: Always start with Direct Substitution

Rem: Always start with Direct Substitution

Method 1: Algebraic - Factorization

4

2 04 0x

xLimx

Method 2: Algebraic - Rationalization

3

1

1 01 0x

xLimx

Method 3: Numeric – Chart (last resort!)3

0

1 00

x

x

eLimx

Method 4: CalculusTo be Learned Later !

Do All Functions have Limits?Where LIMITS fail to exist.

0

1, 03, 0x

xLim

x

2

42x

xLimx

0

1sinxLim

x

Why?

0xLim x

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

Homework Problems

1. From the figure,

determine a

such thato

( ) 2 5, 3, 1, 0.2f x x a L

Review:(5). The graph of the function displays the graph of a function with

Estimate how close x must be to 2 in order to insure that f(x) is within 0.5 of 4.

2( ) 4

xLim f x

(6). Find a such that 0 ( ) (5 3 ), 2, 1, 0.05f x x a L

Last Update:

• 08/12/10

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

xLim 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

aThis theorem allow us to use DIRECT SUBSTIUTION with Trig Functions.

Limits of TRIG Functions:cont.

In a UNIT CIRCLE measured in RADIANS:

sin( )

w PQ

w PN

PQ PQ

THEREFORE:

sin( ) sinx aLim x a

Defn. of radians!

Exponential and Logarithmic Limits

Use DIRECT SUBSTITUTION. REM: the Domain of the functions

x a

x aLime e

ln( ) ln( )

x aLim x a

REM: Special Exponential Limit0

1 1x

x

eLimx

For a > 0