Limit Kuliah Ke4

8/19/2019 Limit Kuliah Ke4

1/38

Teaching Limits so that

Students will Understand Limits

Presented by

Lin McMullinNational Math and Science

Initiative


2/38

( )3 24 6

2

x x x f x

x

− + +=

−

−4 −3 −2 −1 1 2 3 4

−4

−3

−2

−1

1

2

3

4

Continuity

What happens at x = 2?

What is f (2)?

What happens near x = 2?

f ( x ) is near 3

What happens as x approaches 2?

f ( x ) approaches 3


3/38

What happens at x = 1?

What happens near x = 1?

As x approaches 1 g increases !itho"t bo"nd or g approaches in#nity$

As x increases !itho"t bo"nd g approaches %$

As x approaches in#nity g approaches %$

Asymptotes

−4 −3 −2 −1 1 2 3 4 5 6

−1

1

2

3

4

5

6

( )( )

2

1

1 g x

x=

−


4/38

Asymptotes

−4 −3 −2 −1 1 2 3 4 5 6

−1

1

2

3

4

5

6

( )( )

2

1

1 g x

x=

−

& &'1 1(&'1)2

0.9 -0.1 100.00

0.91 -0.09 123.46

0.92 -0.08 156.25

0.93 -0.07 204.08

0.94 -0.06 277.78

0.95 -0.05 400.00

0.96 -0.04 625.00

0.97 -0.03 1,111.11

0.98 -0.02 2,500.00

0.99 -0.01 10,000.00

1 0 Undefned

1.01 0.01 10,000.00

1.02 0.02 2,500.00

1.03 0.03 1,111.11

1.04 0.04 625.00

1.05 0.05 400.00

1.06 0.06 277.78

1.07 0.07 204.08

1.08 0.08 156.25

1.09 0.09 123.46

1.10 0.1 100.00


5/38

Asymptotes

−4 −3 −2 −1 1 2 3 4 5 6

−1

1

2

3

4

5

6

( )( )

2

1

1 g x

x=

−

& &'1 1(&'1)2

1 0 Undefnned

2 1 1

5 4 0.25

10 9 0.01234567901234570

50 49 0.00041649312786339

100 99 0.00010203040506071

500 499 0.00000401604812832

1,000 999 0.00000100200300401

10,000 9999 0.00000001000200030

100,000 99999 0.00000000010000200

1,000,000 999999 0.00000000000100000

10,000,000 9999999 0.00000000000001000

100,000,000 99999999 0.00000000000000010


6/38

1 2 3 4−1

1

2

3

4

5

6

7

8

9

10

The Area Problem

( )

( )

21

0

1

3

h x x

j x

x

x

= +

=

=

=

What is the area o* the o"tlined re+ion?

As the n",ber o* rectan+les increases !ith o"t bo"ndthe area o* the

rectan+les approaches the area o* the re+ion$

1 2 3 4−1

1

2

3

4

5

6

7

8

9

10

1 2 3 4−1

1

2

3

4

5

6

7

8

9

10

1 2 3 4−1

1

2

3

4

5

6

7

8

9

10


7/38

The Tangent Line Problem

What is the slope o* the blac- line?

As the red point approaches the blac- point the redsecant line approaches the blac- tan+ent line and

.he slope o* the secant line approaches the slope o* thetan+ent line$


8/38

As x approaches 1, ! " #x $ approaches %

f ( x ) wi t h i n 0 . 0 8

uni t s

o

f 3 xwi t h i n 0 . 0 4 un

i t s

of 1

f (

x ) wi t h i n 0 .1 6 u

ni t s

of 3

x

wi t h i n 0 . 0 8 uni t s

o

f 1

0.90 3.20

0.91 3.18

0.92 3.16

0.93 3.14

0.94 3.12

0.95 3.10

0.96 3.08

0.97 3.06

0.98 3.04

0.99 3.02

1.00 3.00

1.01 2.98

1.02 2.96

1.03 2.94

1.04 2.92

1.05 2.90

1.06 2.88

1.07 2.86

1.08 2.84

1.09 2.82

x 5 2 x−


9/38

( )→

− =1

lim 5 2 3x

x

( )

( )

12

2 2

2 2

5 2 3

5 2 3

x

x

x

x

x

f x L

ε

ε

ε

ε

ε

ε

− <

− <

− <

− − <

− − <

− <


10/38

−2 −1 1 2 3

−1

1

2

3

4

5

1 or 1 12 2 2

x xε ε ε

− < − < < +

( )5 2 3

or

3 5 2 3

x

x

ε

ε ε

− − <

− < − < +

( )→

− =1

lim 5 2 3x

x

Gr!h


11/38

( )→

− =1

lim 5 2 3x

x

2

ε δ = δ ε =

4

ε δ =


12/38

"hen the #$ues su%%essi#e$&ttri'uted to #ri'$e !!ro%h

indefnite$& to fed #$ue, in nner so s to end '& di*erin+ro it s $itt$e s one wishes,

this $st is %$$ed the $iit o $$the others.

u+ustin-ouis

u%h& (1789 / 1857)

The &e'inition o' Limit at a Point


13/38

( )

( )

( ) ( )

lim if, and only if, for any number 0

there is a number 0 such

if 0 ,

t

then

hat

x a

x

L

f x L

f

a

x ε

ε

ε

ε

δ

δ

→

< −

>

< −

=

<

>


r$ "eierstrss

(1815 / 1897)


14/38

( )

( )

( ) ( )


there is a number 0 such

if 0 ,

t

then

hat

x a

x

L

f x L

f

a

x ε

ε

ε

ε

δ

δ

→

< −

>

< −

=

<

>

( )

( )( ) ( ) ( )


there is a numb

i

er 0 and such t

f , then

hat

x a

a x a L f x L

f x L

x a

δ ε δ

ε

ε

δ ε

ε ε

→= >

− < < + − < <

≠

+

>


r$ "eierstrss

(1815 / 1897)


15/38

( )( )

lim 0 0 such that

, whenever 0

x a f x L

f x L x a

ε δ

ε δ

→ = ⇔ ∀ > ∃ >

− < < − <

(ootnote)The &e'inition o' Limit at a Point


16/38

(ootnote)The &e'inition o' Limit at a Point

( )( )

lim 0 0 such

whe

that

, 0never

x a f x L

f x L x a

ε δ

ε δ

→ ⇔ ∀ ∃= > >

− < < − <


17/38

f

( x ) wi t h i n 0 . 0 8

uni t s

o

f 3

xwi t h i n 0 . 0 4 un

i t s

of 1

f (

x ) wi t h i n 0 .1 6 u

ni t s

of 3

x

wi t h i n 0 . 0 8 uni

t s

o

f 1

0.90 3.20

0.91 3.18

0.92 3.16

0.93 3.14

0.94 3.12

0.95 3.10

0.96 3.08

0.97 3.06

0.98 3.04

0.99 3.02

1.00 3.00

1.01 2.98

1.02 2.96

1.03 2.94

1.04 2.92

1.05 2.90

1.06 2.88

1.07 2.86

1.08 2.84

1.09 2.82

x 5 2 x−

( )→

− =1

lim 5 2 3x

x


18/38

( )→

− =1

lim 5 2 3x

x

( )

( )5 2 3

5 2 3

2 2

2 2

12

f x L

x

x

x

x

x

ε

ε

ε

ε

ε

ε

− <

− − <

− − <

− <

− <

− <


19/38

→

=2

3lim 9x

x

2 9

3 3

x

x x

ε

ε

− <

+ − <


20/38

→

=2

3lim 9x

x

2 9

3 3

x

x x

ε

ε

− <

+ − <

Near 3, specifically in 2,4!, 5 3 " x x= < + <


21/38

→

=2

3lim 9x

x

2 9

3 3

x

x x

ε

ε

− <

+ − <


" 3

3 "

x

x

ε

ε

− <

− <


22/38

→

=2

3lim 9x

x

2 9

3 3

x

x x

ε

ε

− <

+ − <


" 3

3 "

x

x

ε

ε

− <

− <

( ) the smaller of 1 and"

ε δ ε = Gr!h


23/38

→

=lim sin( ) sin( )x a

x a

Gr!h1

1

a + delta

a - delta

sin x

sin a

(cos x, sin x)

(cos a, sin a)

1, 0!

x in radians

0,1!


24/38

( )

( )

( ) ( )

( )

( )

( )


there is a number 0 such that

if 0 , then


there is a number 0 such that

if 0 , t

a

a

x

x

f x L

f x x a L

f x L

a x

ε

δ ε

δ ε ε

ε

δ ε

δ ε

+

−

→

→

−

= >

>

< <

−

− <

= >

>

< < ( )hen f x L ε − <

*ne+sided Limits


25/38

Limits -ual to .n'inity

( )

( )

( ) ( ) ( )( )

lim if, and only if, for any number

there is a number such that

if

0

/raphically this is a vertical asymptote

0 , then

x a M

f x M M

f x

x a f x

δ ε

δ ε

→ =

<

>

( )( )

( ) ( )

lim if, and only if, for any number

there is a number such that

if 0 , then

0 x a f x

x

M

f xa M

δ ε

δ ε

→ =

< − <

−∞ <

<


26/38

Limit as x Approaches .n'inity

( )

( )


there is a number suc

/raphically, this is a horiontal a

h that

if , then

sym t

0

p ote

x f x L

M f x L

M

x

ε

ε

∞→= >

−> <

>

( )

( )

lim if, and only if, for any number 0there is a number such that

if , then

0

x

M

x

f x L

f x L M

ε

ε

→−∞

<

<

= >

− <


27/38

Limit Theorems

$ost $$ $iit re %tu$$& ound '&su'stitutin+ the #$ues into the e!ression,si!$i&in+, nd %oin+ u! with nu'er,

the $iit.

he theores on $iits o sus, !rodu%ts,!owers, et%. usti& the su'stitutin+.

hose tht dont si!$i& %n oten 'e oundwith ore d#n%ed theores su%h s!it$s u$e


28/38

The Area Problem

1 2 3 4−1

1

2

3

4

5

6

7

8

9

10

( )

( )

21

0

1

3

h x x

j x

x

x

= +

=

=

=


29/38

The Area Problem

1 2 3 4−1

1

2

3

4

5

6

7

8

9

10

( ) ( )

( ) ( ) ( ){ }

( )( ) ( )

( )( ) ( )

2

2 2 2 2

22 2

1

2

2 2

1

len$th 1

3 1 2width

)*coordinates ' 1,1 ,1 2 ,1 3 , ,1

rea 1 1

32rea ' lim 1 1 3

n n n n

n

n n

i

n

n nn

i

h x j x x

b a

n n n

n

i

i

=

→∞=

= − = +

− −= = =

+ + + +

≈ + +

+ + =

∑

∑


30/38

The Area Problem

1 2 3 4−1

1

2

3

4

5

6

7

8

9

10

( )( ) ( )

( ) ( ) ( )

2

2 3

2 3

2 22 2 2 2 4 2 4

1 1 1 1

4 2

1

1 2 1

6

1

4

3

32

1

3

2

1

lim 1 1 lim 2 lim lim

lim lim lim

lim lim lim

4 4

1

n n n n

n n n n n n ni n n ni i i i

n n nn n n

n n n

n

i

n

n n n

n

n n

i

n

n

n

i

i

i i

n

i

i

→∞ →∞ →∞ →∞= = = =

→∞ →∞ →∞

→∞ →∞ →

=

+

==

+

∞

+

+ + = + +

= + +

= + +

= + +

=

∑∑

∑

∑

∑ ∑ ∑


31/38

( )( ), P a x f a x+ ∆ + ∆

( )( ),T a f a

x∆

( ) ( ) f a x f a+ ∆ −

( ) ( ) y f a x am= + −


( ) ( )

( )

s

0

slope

P T

x

PT

f a x f a

a x a

m

m

→

∆ →

→

+ ∆ −→

+ ∆ −


32/38

( )( ), P a x f a x+ ∆ + ∆

( )( ),T a f a

x∆

( ) ( ) f a x f a+ ∆ −

( ) ( ) ( )a y f a x a f ′= + −


( ) ( )

( ) ( )

s

0

slope m

P T

x

PT

f f

a x f a

a xa

a

→

∆ →

→

−→

+ ∆ −′

+ ∆


33/38


34/38

in %u$$intion$ th nd :%ien%e;nititi#e

325 orth :t.


35/38

0in McM"llin

l,c,"llin/NationalMathAndScience$or+

!!!$0inMcM"llin$net lic- AP


36/38

( ) ( )

( ) ( )

( ) ( )

#et

sin sin

sin sin

limsin sin x a

x a x a

x a x a

x a

δ ε

δ ε

→

=

− < −

− < ⇒ − <

⇒ =


37/38

−2 −1 1 2 3

−1

1

2

3

4

5

( ) x a δ ε − <

( ) f x L ε − <

( )→

− =1

lim 5 2 3x

x

( )

( )

( ) ( )

lim if, and only if,

for any number 0 there is a

number 0 such that if

0 , then

x a f x L

x a f x L

ε

δ ε

δ ε ε

→=

>

>

< − < − <


38/38

( )( ), P a x f a x+ ∆ + ∆

( )( ),T a f a

x∆

( ) ( ) f a x f a+ ∆ −

( ) ( ) y f a x am= + −


( ) ( )

( )

s

0

slope

P T

x

PT

f a x f a

a x a

m

m

→

∆ →

→

+ ∆ −→

+ ∆ −

Limit Kuliah Ke4

Documents