Limit Fungsi Genap1

8/19/2019 Limit Fungsi Genap1

1/64

March 18, 2016 Editor Hendry. P 1

1

PENDAHULUAN

2

PEMBAHASAN

3

PENUUP


2/64

LOGO


STANDAR KOMPETENSI

MEN!!UNA"AN "#NSEP L$M$

%UN!S$ DAN U&UNAN %UN!S$

DALAM PEME'AHAN MASALAH


3/64


"#MPEENS$ DASA&

1.MEN(ELAS"AN SE'A&A $NU$$%

A&$ L$M$ %UN!S$ D$SUAU $$"DAN D$ A" H$N!!A

2.MEN!!UNA"AN S$%A L$M$ %UN!S$

UNU" MEN!H$UN! BENU" A"ENU %UN!S$ AL(ABA& DAN

&$!#N#ME&$


4/64

March 18, 2016 Editor Hendry. P )


5/64

March 18, 2016 Editor Hendry. P *

SEELAH MEMPELA(A&$ BAB $N$,PESE&A D$D$" D$HA&AP"AN

MAMPU MEN(ELAS"AN SE'A&A

$NU$$% A&$ L$M$ %UN!S$

D$SUAU $$" DAN D$ A"

H$N!!A ,MEN!!UNA"AN S$%A

L$M$ %UN!S$ UNU"

MEN!H$UN! BENU" A" ENU%UN!S$ AL(ABA& DAN

&$!#N#ME&$


6/64

LOGO

March 18, 2016 Editor Hendry. P 6Materi Kelas XI Semester 4


7/64March 18, 2016 Editor Hendry. P +

L$M$ %UN!S$ AL(ABA&1

E#&EMA L$M$2

L$M$ %UN!S$ &$!#N#ME&$3

DAL$L LH#SP$AL4


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Limit = Pendekatan

Berarti Limit Fungsi = Pendekatan Nilai Fungsi

Manfaatnya :

Untuk mentukan nilai fungsi yang memiliki nilai tak

Tentu seperti :

[ ] [ ]∞∞∞

∞∞

x0,-,,0

0


9/64

March 18, 2016 Editor Hendry. P -

Daat di/ee/aian denan f(x)Limitax→

)(Limitx

x f ∞→

A. Bent4

1. S45/tit4/i

2. Pe7atoran

3. Peraian /eaan

B. Bent4 Daat di/ee/aian denan

1. Pe5aian anat tertini enye54t2. Peraian /eaan


10/64


)(Limitax

x f →

25 E.

24 D.

23 C.

22 B.

21 A.

...)]13)(1[(LimitHitungla 22x

=−+→

x x

BENU"


11/64


25

5x5

1)-2.3)(1(2)]13)(1[(xLimit

)]13)(1[(LimitHitungla

22

2x

2

2x

=

=

=+=−+

=−+

→

→

x

x x

#P$#N


12/64


∞

=

+

−

→

E.

2 D.

1 C.

2

1 B.

0 A.

...Limit0x

x x

x x


13/64


101

01

1

1

Limit

)1(

)1(LimitLimit

0x

0x0x

=+

−

=+

−

=

+

−=

+

−

→

→→

x

x

x x

x x

x x

x x

#P$#N


14/64

March 18, 2016 Editor Hendry. P 1)

22 E.

2 D.

22

1

C.

2

1 B.

2

4

1 A.

.....22

Limit0x

=−−+

→ x

x x


15/64

March 18, 2016 Editor Hendry. P 1*

2

2

1

2

1

22

2

0202

2

22

2

Limit

)22(

2

)22(

)2()2(Limit

22

2x2 x

22Limit

0x

0

0x

==

=−++=−++

=−++

=−++

−−+

−++

−++−−+

→

→

→

x x

x x x

x

x x x

x x

x x

x

x

x x

x

#P$#N


16/64


)(LimitBE!"#$x

x f ∞→

0g(x)

f(x)Limit

ma%ag(x)&'aatf(x)&'aat*i%a 3.

-g(x)

f(x)Limit

ma%an'gatif, +'nilaif(x)inggi t't

ang%at%'fii'n i%a'+ali%n/ag(x)

f(x) Limit

ma%a itif, +'nilaif(x)inggi t't

ang%at%'fii'n&ang(x)&'aatf(x)&'aat*i%a .2

g(x)&aitinggi ang%at t'%'fii'n

f(x)&aitinggi ang%at t'%'fii'n

g(x)

f(x) Limit

ma%ag(x)&'aatf(x)&'aat*i%a 1.

/aituaa3&'nganan&i'l'ai% &aat)(

)(Limit

x

x

x

x

x

=

<

∞=

∞=

>

=

=

∞→

∞→

∞→

∞→

∞→ x g

x f


17/64

March 18, 2016 Editor Hendry. P 1+

3 E.

2 D.

1- C.

2- B.

3- A.

...432

34LimitHitungla

23

23

x=

+++−

++−

∞→ x x x

x x x


18/64


19/64

March 18, 2016 Editor Hendry. P 1-

1 E.

2

1 D.

0 C.

B. - A.

...2

132Limit

2

x

∞∞

=+

−+∞→ x

x x


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"arena Dera9at %:;< = dera9at !:;< dan "oe7i/ien anat tertini %:;<

5erniai Po/iti7, aa

∞=

+

−+

∞→

2

132Limit

2

x x

x x

#P$#N


21/64


1 E. D.

0 C.

1- B.

- A.

...24

4234Limit

2

32

x

∞

∞

=

−+

−+−

∞→ x x

x x x


22/64


∞=

−+

−+−

∞→

-244234Limit

2

32

x x x x x x

#P$#N


23/64


1 E.

0 D.

C.

1- B. - A.

...53

143Limit

24

23

x

∞

∞

=++−

++−∞→ x x x

x x x


24/64


"arena dera9at 7:;< > dera9at :;< aa

053

143Limit

24

23

x=

++−

++−

∞→ x x x

x x x

#P$#N


25/64


5312Limit 1.

ini +'i%utfungiLimitHitungla

x+−−

∞→ x x

Lihat Haaan 5eri4t


26/64


6 x

6 x

+ax

+ax x6 x +axLimit i%a*a&i

'%a7an '%alian&'ngan&i%ali%anauitu8al

9x +

+−

+

++++

→


27/64


1 E.

0 D.

9 C.

1- B.

9- A.

5312Limit x

+−−∞→

x x


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9-5312

Limit

5312

)53()12(Limit

5312

5312x x5312Limit

9x

9x

9x

=

++−

−−

=++−+−−

=++−

++−+−−

→

→

→

x x

x

x x

x x

x x

x x x

#P$#N


29/64


)3)(1(Limit .2

1123Limit .1

x

22

x

x x x

x x x x

−++

+−−+−

∞→

∞→

Lihat Haaan Beri4t


30/64


9-limitn/ama%a ,a*i%a 3.

9limitn/ama%a ,a*i%a .2

a2

6- + limitn/ama%a ,a*i%a 1.

ma%a) 6x x +xax(Limit 22x

=<=>

==

+−−++∞→


31/64


1 E.

31 D.

0 C.

9 B.

9- A.

1123Limit .1 22x

+−−+−∞→

x x x x


32/64


33/64


2 E.

1 D.

0 C.

2

1 B.

2

1- A.

...)3)(1(Limitx

=−++∞→

x x x


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22

4

12

0-4

a2

6- +

Limitn/a ma%a ,a $a'na

=

==

=

#P$#N


35/64


)5434x(Limit Hitungla 22x

x x x −−+∞→

3 E.

2 D.

1 C.

0 B.9 A.


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2

4

42

)5(3a2

6- +a&alalimitn/ama%a a$a'na

==−−

=

#P$#N


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0 E.

2

1 D.

4

5 C.

54 B.

9 A.

...345)x(4xLimit 2x

=−−+∞→

x


38/64


4

5

42

05a2

6- +a&alalimitn/ama%a ,a $a'na

=−

=

#P$#N


39/64


2 E.

1 D.2

1 C.

0 B.

9 A.

...)2x-(xLimit 2x

=−∞→

x


40/64

March 18, 2016 Editor Hendry. P )0

1

2

2

12

20a2

6- +a&alalimitn/ama%a ,a $a'na

==+

=

#P$#N


41/64


0g(x)Limit:g(x)Limit

f(x)Limit

g(x)

f(x)Limit .

f(x)Limit%%(f(x))Limit 5.

g(x)Limitf(x).Limitf(x).g(x)Limit 4.

g(x)Limitf(x)Limitg(x)f(x)Limit 3.

af(x)Limitma%ax,f(x)*i%a 2.

f(x)Limitma%a,f(x)*i%a 1.

ax

ax

ax

axax

axaxax

axaxax

ax

ax

≠=

=

=+=+

==

==

→

→

→

→

→→

→→→

→→→

→

→

a x


42/64


PENE&APAN E#&EMA L$M$ UAMA

...;

Limit .

...)5(4xLimit +.

...3Limit a.

Hitungla

23x

2

4

2

x

=+

=+

=

→

→

→

x

x

x

x

x


43/64


1


44/64

March 18, 2016 Editor Hendry. P ))

[ ][ ] [ ]

4

204

4544

xLimit5Limit4

xLimit5xLimit4

5xLimit4xLimit

)5(4xLimit +.

2

4x

2

4x

4x

2

4x

4x

2

4x

2

4

=

+=

+

+

+

+

=+

→→

→→

→→

→

x

x x


45/64

March 18, 2016 Editor Hendry. P )*

3

;

Limit 1.23x

=

+→

x

x

;LimitxLimit3x

2

3x →→

+

4

3

;3

3

2

=

+

=


46/64


[ ]

[ ]f(x).g(x)Limitn

&'ngan'f(x)1Limit 2.

g(x).f(x)Limitm

&'ngan'f(x)1Limit .1

0x

ng(x)

0x

x

mg(x)

x

→

→

∞→

∞→

==+

=

=+


47/64

March 18, 2016 Editor Hendry. P )+

[ ] '1Limit .3

'

1

1limit .2

'

1

1Limit 1.

1

0x

x

x

=+

=

−

=

+

→

−

∞→

∞→

x x

x

x

x

x


48/64


' E

' D.' C.

0 B.

'

1 A.

....2

11Limit

2

9n

312

=

−

++++

→

nn

n

n


49/64

March 18, 2016 Editor Hendry. P )-

2

9n

2

2

2

9n

9n

9

m

9n

'2

11Limit *a&i

2m nilai&i'l'

n&i+agi 'n/'+ut&an='m+ilang

32Limitm

3n12n.

2-n1n Limitm

3n

12n g(x) :

2-n

1n f(x)

f(x).g(x)Limitm ana&im,2'2n

1n1Limit

312

=

−

++

=

−+

+=

++

+=

+

+=

+=

==

−

++

→

→

→

→→

++

n

n

nn

nn

nn

nn

#P$#N


50/64

March 18, 2016 Editor Hendry. P *0

2

4

29x

' E.

' D.'

1 C.

1 B.0 A.

....1

1Limit

2

=

+

−

→

x x

x


51/64


[ ]

'1

1Limit *a&i

1m

&i&aat&'ngan x&i+agi 'n/'+ut&an='m+ilang

4x Limit4x

x

1 Limitm

*a&i 4xg(x)&anx

1f(x)

f(x).g(x)Limitm&imana'1

1Limit

4

29x

2

2

2

9x

2

29x

2

2

9x

m

4

29x

2

2

=

+

=

−=−

=

−==

==

+

−

→

→→

→

−

→

x x

x x

x

x

x

x

#P$#N

SOAL LATIHAN


52/64

SOAL LATIHAN


)1(Limit . )11(Limit .4

)1(Limit .; )3

21(Limit .3

)1(Limit . )2

1

1(Limit .2

)21(Limit .5 )x

1(1Limit 1.

ini +'i%utlimit-LimitHitungla

1

0 x

429 x

1

0 x9 x

2

0 x9x

1

0 x

1x

9x

2 x x x

x x

x x

x

x x

x x

x x

x

−+

−−

−+

++

→−

→

→

−

→

→→

→

+

→


53/64



54/64

March 18, 2016 Editor Hendry. P *)


55/64

March 18, 2016 Editor Hendry. P **


56/64



57/64

March 18, 2016 Editor Hendry. P *+

1 E.

21 D.

0 C.

9 B.

9- A.

.....2x8in

8in xLimit

0x=

→


58/64


2

1

2.1

1

x2C

1Limit

xC28in x

8in xLimit

2x8in

8in xLimit

0x

0x0x

=

=

=

=

→

→→

#P$#N


59/64

March 18, 2016 Editor Hendry. P *-

1 E.

21 D.

0 C.

9 B.

9- A.

...8in x

2xC1Limit

0x=−

→


60/64


0 08in2

8in x2Limit

8in x

x8in2 Limit

8in x

2xC1Limit

0x

2

0x0x

==

=

=−

→

→→

#P$#N


61/64


1 E.

0 D.

1- C.

9 B.

9- A.

...

)

3

(x"an

)3

>(x

Limit3

x=

+

+

−→ π π


62/64


1

)3

(x"an

)3

>(x

Limit*a&i

1u"an

ulimit

)3

(x"an

)3

>(x

Limit'ingga

0,uma%a,3-x*i%a3 xu?ial%an

)3

(x"an

)3

>(x

Limit

3

3

3

-x

0u-x

x

=+

+

==+

+

→→+=

+

+

→

→→

−→

π

π

π π

π

π

π

π

#P$#N


63/64


g(x)&anf(x)fungintuuna

a&ala(x)@g&an(x)@f (x)@g

(x)@f Limitg(x)

f(x)Limit

8++ H8="ALL@ Dalil

nm'ngguna%a&'nganan&i'l'ai% &aat'+utLimit t'

ma%a,0

0anm'ngail% f(x) a&axnilai%anu+titui%ita i%a

i"ignm'tfungiLimitmauunAla+a,fungiLimitAa+ila

axax →→ =

LOGO


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Limit Fungsi Genap1

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