Limit Calculus)

8/8/2019 Limit Calculus)

1/30

Limits

Limits are used to describe how a function behave as the independent variable move toward

a certain value. Consider a funtion 1,111)(

2{!

! xx

xxxf and the tables below .

Table Table

l)(l

!

pp x

xxf

xx

!

pp xxxf

xx

2

1lim

1

11lim

1

1

!

!

!

p

p

x

x

xx

x

x

2

1lim

1

11lim

1

1

!

!

!

p

p

x

x

xx

x

x

Since lili !!

pp

xfxfxx

, therefore !p

xx

.

The function is undefined at !x but the limit does exist.

x xf

. .

.0 .0

,00 .00

.000 .000

.0000 .0000

x xf

0.9 .9

0.99 .99

0.999 .999

0.9999 .9999

0.99999 .99999


2/30


3/30

Example

!

"

!

3if7

3if4

3if4

xx

x

xx

xf

27lim)(lim 233

!! ppxxf

xx

242lim)(lim33

!!

pp

xxfxx

!

p

xfx

Example

lim)(lim00

!!

pp

xxfxx

g!!

pp xxf

xx

2li)(li

00

istnotoes)(li0

xfxp

y

x

y

x

e

"!

0if

0if)(

xx

xxxf


4/30

Example

!x

x

g!

!

p

p

p 3li

li

li

3

3

3 x

xf

x

x

x

!

!

p

p

p 3lim

lim

lim

3

3

3 xxf

x

x

x

istnotoes)(li xfx p

Example

!

xxf

g!

!

p

p

p 2

2

2

2

2l

1l

l

x

xf

x

x

x

g!

!

p

p

p2

2

2

2 2lim

1lim

)(limx

xf

x

x

x

g!

)(li2

xfx

y

x-3

y

x2


5/30

Exa pl { l xfx g

and l xfx g

}

1l m !g

xfx

4)(lim

!gpx

fx

Exa pl { l xfx g and l xfx g }

6)(lim !gp

xfx

g!gp

)(lim xfx

-1

y

x

4

-3

y

x

6


6/30

Exa pl { )(li xfx gp

and )(li xfx gp

}

g

gp

)(li xfx

g!gp

)(lim xx

L t Co putat onal t chn qu s

kkcx!

p

lim , wh r k sa constant.

kkkxx

!!gpgp

limlim

cxcx!

lim

-3

y

x2 6


7/30

Example 3lim3

!

xx

0li0

!

xx

2li2

p

xx

g!gxxli g!

gx

xlim

g!p xx

1l g!p xx

1l xistnotdoes1lim0 xxp

i3

!

xx

2

11li2

! xx

0lim !p xx

0lim !!p xx


8/30

Theorem

Let m stand for oneof the m ts

c"

p

li ,

pcx

lim ,

pcx

lim ,

gpx

lim or

gpx

lim . If )(lim Lxf ! and

2li Lxf ! bothexist, then

a limlimlim LLxgxfxgxf !! .

b (i(i((i LLxgxfxgxf !! .

c 21)(i)(i)()(i LLxgxfxgxf !! .

d if,(li

(li

(

(li 2

2

1 {!!

L

L

L

xg

xf

xg

xf.

e eveinifovi e(li(li 11 "!! LLxfxfnnn .


9/30

Example

a)

03)3(33limlim4lim34lim

2

33

2

3

2

3

!!! pppp xxxx

xxxx

b)

8

24

4)4(5

2li

45li

245

li

4

4

4!

!

!

p

p

p x

x

xx

x

x

x

c) 42li

2

22li

24li

22

2

2!!

!

ppp

xx

xx

xx

xxx

d)

exist.otdoes12

1li

2

p xxx

x

g!

# 12

1li

2 xx

x

x

g!

$ 12

1li

2 xx

x

x


10/30

E mp Find

a)28

li3

2

p xx

x b)

11li

p x

x

x

c) )(li xfxp

if

u!

3if2)1(3

3if13)(

2xx

xxxf

Solution

a) 1222lim

2

222lim

2

8lim 22

2

22

2

%

2

!!

!

&

&

&

xxx

xxx

x

x

xxx

b) 211

li11

11

11li

11li

000!!

v

! ppp x

xx

x

x

x

x

x

x

xxx

c) 11)()1(li)(li33

!!!

''

xxfxx

102)2(32)1(3lim)(lim22

33 !!! (( xxf xx

10li3

!@)

xfx


11/30

Example Find

a) )(li2

xfx p

if

"

e!

2if134

2if32)(

2 xxx

xxxf

b) a if

"

e!

2if2

2if1-)(

2

3

xax

xxxf and )(lim

2xf

xpexists.

Solution a) 91)2(3)2(4)134(lim)(lim22

22!!!

ppxxxf

xx

228)2(3232lim)(lim 22 !!!! pp xxf xx )(lim

2xf

x p@ does not exist.

b) )(lim2

xfxp

exists )(lim)(lim22

xfxfxx

00

!

4

5

54

1)2(2)2(

)1(lim)2(lim

32

3

2

2

2

!

!

!

! 11

a

a

a

xax xx


12/30

Limitsfortrigonometricfunctions

Letcis real number

a) cxcx

sinsinlim !p

b) cxcx

coscosli !p

c) cxcx

tantanli !p

d) cxcx

csccsclim !p

e) cxcx

secseclim !p

f) cxcx

cotcotli !p

ample

a) 12

iili

2

!T!T

2

x

x

b)4

4

14

1

14

4tan

1tanli

4

T!

T

!

T

T

!T

px

x

x

c) 2

4cos

1cos

1limcossin

coscossin

limcossin

1tanlim

444

!T

!!

!

T3

T3

T3

xxxx

xx

xxx

xxx


13/30

Speci l Li its

I. 1silim !p x

x

x

II. 0cos1lim0

!p x

x

x

Example

a

)1

cos

1

li

sin

li

cos

sin

li

tan

li !!!

pppp xx

x

xx

x

x

x

xxxx

b) 44

4sinlim44

44sinlim4sinlim !!v!444

xx

xx

xx

xxx

c) 0cos1limsilim)cos1(si

limsicossi

lim00

20

20

!!

!

pppp x

x

x

x

x

xx

x

xxx

xxxx


14/30

Example (Using 01liand01li !!gpgp xx xx

)

a)6

3

68

53lim

68

53lim !

!

5p5p

x

xx

x

xx b) !

!

6p6p

x

xx

xx

xx 11

23

lim1

23lim

34

c) 052

14

li52

4li

2

2

3

2

!

!

g7g7

x

xx

x

xx

xx d) g!

!

gpgp

x

xx

x

xx 11

23lim1

23lim34

.


15/30

Definition

A line cx! iscalled a verticalasymptoteofafunction )(xf if 89

)(xf or gp)(xf as

@ cx or p cx .

A line Ly ! iscalled a horizontelasymptoteofafunction )(xf if Lxf p)( as gpx or

gpx .

Theorem

A line cx! isa verticalasymptoteofarationalfuntionxq

xpxf !)( ,if cx isafoctorofthe

denominator )(xq and notacommonfactorfor )(xp and )(xq .


16/30

Example Findthe ve ticalandhorizontalasymptotes i any orthe ollowin unctions.

a)13)(

!

xxxf b)

4

)(2

!

x

xxf

c)4

8)(2

3

!x

xxf d)1

1)(23

!xxx

xxf

Solution

a) 1!x is the verticalasymptote.

11

1

31

lim1

3lim !

!

A

pA

p

x

xxx

xx. 1!@ y is the horizontalastmptote

b) 224

)(2

!

!xx

x

x

xxf , 2!x and 2!x are the verticalasymptotes.

041

1

lim4

lim

2

2!

!

gpgp

x

x

x

x

xx. 0!@ y is the horizontalastmptote


17/30

c)

22228)(

2

2

!

!

xx

xxx

x

xxf 2!x istheverticalasymptote.

g!

!

gpgp

2

2

2

3

11

8

lim4

8lim

x

xx

x

xxx

. )(xf@ has no horizontalastmptotes

d) 1)1(1

1)1(

1

1

1)(2223

!

!

!

xx

x

xxx

x

xxx

xxf

)(xf@ has no vertical astmptote

01111

11

li1

1li

32

32

23!

!

gpgp

xxx

xx

xxx

x

xx.

0!@ y isthe horizontalastmptote


18/30

Example Fi d a)63

2lim

2

gp xx

x b)

63

2lim

2

BC xx

x

c)thehorizontaland verticalasymptotesi any for .62)(

2

!

x

xxf

Note : xx !2

for 0"x and xx !2

for .0x

Sol tion

a) 31

63

21lim

/63/2

lim63

2lim

2222

!

!

!

D

E

D

E

D

E

x

xxxxx

xx

xxx

b) 31

63

21

lim/63/2

lim63

2lim

2222

!

!

!

F

G

F

GFG

x

xxxxx

xx

xxx

c) From(a)an(b),31

!y and3

!y arethehorizontalasymptotesand 2!x isthe vertical

asymptote for .62

)(2

!x

xxf


19/30

Definition

A function )(xf issaid to becontinuousat cx! ifthefollowing conditionsaresatisfied.

a) )(cf is defined.

b) )(lim xfcxp

exists

c) )()(lim cfxfcx

!p


20/30

Example Giventhe graph of )(xf .

Determine whether )(xf continuousat cx ! for 6,4,1!c .

Solution

At 1!x , 3)1( !f and )(lim1

xfx p

doesnotexist.

)(xf@ is discontinuousat 1!x .

At 4!x , 5)4( !f and 3)(li4

!

p

xfx

.


At 6!x , 3)6( !f and 3)(lim6

!

p

xf

x

.

)(xf@ iscontinuousat 6!x .

5

3

-1

-2

-3

6

4

y

x


21/30

Exa ple iven

"!

e

e

!

5if221

5if5

52if133

23if4

3if2

13

)( 2

xx

x

xx

xx

xxx

xf .

Deter ine whether )(xf continuousat cx! for 5,2,!c .

Solution At 3!x , 1)( !f and g!!

pp 6213lim)(lim

33 xxxf

xx. )(lim

3xf

x H doesnotexist.


At 2!x , 21)32(3)2( 2 !!f , 264lim)(lim22

!!

ppxxf

xx

,

21)3(3lim)(lim 222

!!

II

xxfxx

2)(li2

!p

xfx

. )(xf@ is continuousat 2!x .

At 5!x , 5)5( !f ,

111)3(3lim)(lim 255

!!

PP

xxfxx

, .11221lim)(lim55

!!

pp

xxfxx

)5(11)(li2

fxfx

{!

p. )(xf@ is discontinuousat 5!x .


22/30

Derivative of a function

romthe diagramx

Gradiaenth

xfhxfAB

)()( !

Gradiaentofthe curve ath

xfhxfA

h

)()(li

0

!

p

If ),(xfy ! then

.)()(

li)(

d

d

0

,

h

xfhxfxf

x

y

h

!!

Q

This process is known asdifferentiation fromthe first principlesor differentiation using definition.

)(xf

y

x

)( hxf

x x

B

A

)()( xfhxf


23/30

Example Findthederivativeof2

1)(x

xf ! withrespecttoxfromthefirstprinciples.

Solutionh

xfhxfxf

x

y

h

)()(lim)(

0

, !!p

34

220

22

2

0

22

222

0

22

22

0

22

0

22

2li

2li

2li

li

11

li

xx

x

xhx

hx

xhxh

hxh

xhxh

hxhxx

xhxh

hxx

h

xhx

h

h

h

h

h

!!

!

!

!

!

!

p

p

p

p

p


24/30

Example

Given 22)( 2 ! xxxf .Find )(' xf fromthe firstprinciples.

Solutionh

xfhxfxf

h

)()(lim)(

0

, !

R

14

124lim

24lim

222242lim

2222lim

0

2

0

222

0

22

0

!

!

!

!

!

S

S

S

S

x

x

x

xxxxx

xxxx

T

T

T

T


25/30

Example

Given 21

)( xxf ! .Find )(' xf fromthefirstprinciples.

Solutionh

xfhxfxf

h

)()(lim)(

0

, !

U

x

xhx

xhxh

h

xhxh xh

x

xhx

xhxh

xhx

h

h

h

h

2

1

1li

li

li

li

0

0

0

0

!

!

!

!

v

!

p

p

p

p


26/30

E a ple Show that .sis xxx

!

Solution )('cosdd

xfxx ! for .cos)( xxf !

Fro the first principles

h

xfhxfxf

h

)()(lim)(

0

, !

V

xxx

hx

hx

hx

h

xh

x

h

x

h

xx

hxxx

h

xhx

hh

hh

h

h

h

h

sin)1(sin)(c s

sinhlimsin1c sh

limc s

sinhsinlim1c shc s

lim

sinhsin1c shc slim

sinhsin1c shc slim

c ssinhsinc shc slim

c s)c s(

lim

!!

!

!

!

!

!

!

pp

pp

p

p

p

p


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Limit Calculus)