LEMBAR KEGIATAN SISWA (LKS) 1 Nama: .................................................. Kelas: .................................................. No Absen: ..................................................
A. LIMI LIMIT T FUNG FUNGSI SI ALJA ALJABA BAR R Kalkulus merupakan salah satu cabang matematika yang mempunyai peranan penting dalam kehidupan, misalnya menghitung isi badan pesawat, dan menentukan laju perubahan kecepatan pada mobil. Pada bab ini, kita akan mempelajari pokok bahasan limit fungsi. Lalu apa hubungan limit fungsi dan kalkulus? Limit merupakan teori yang mendasari kalkulus, sehingga untuk selanjutnya limit selalu digunakan pada bagian-bagian lain dari kalkulus Pada lembar kerja siswa berikut ini didesain kegiatan siswa yang melalui tahapan Think --> Pair Share. ujuan dari tahapan-tahapan tersebut adalah siswa dapat membangun sebuah konsep matematika secara benar sehingga siswa pada akhirnya dapat melakukan kegiatan menganalisa, mencoba, menyimpulkan dan pada akhirnya dapat membuat keputusan dengan tepat dan akurat serta dapat menge!aluasi.
A.1. Pengertian Limit Fungsi Aljabar Tujuan Pembelajaran: Peserta didik dapat memahami dan menghitung limit fungsi aljabar di suatu titik secara intuitif.
Tahap Think
1. "ermatilah fungsi berikut ini
f ( ( x ) = x + 1
a. "oba tentukan nilai fungsi f ketika # •
$ilai x mendekati % dari sebelah kiri, dengan mengisi tabel dibawah ini. x
1 ,6
1,7
& ,'
&,(
& ,( (
& ,( ( (
& ,( ( ( (
& ,( ( ( ( (
...
→ %
f ( ( x ) ......
......
. . . . ..
......
......
......
......
......
...
→ .. ......
•
$ilai x mendekati % dari sebelah kanan, dengan mengisi tabel dibawah ini. x
2,4
2,3
% ,%
%,&
% ,) &
% ,) ) &
% ,) ) ) &
% ,) ) ) ) &
...
→ %
*asalah Pertama
( x ) ...... f (
......
. . . . ..
......
......
......
......
......
...
→ .. ......
•
+erdasarkan temuan kalian pada tabel, bagaimanakah nilai fungsi mendekati % dari sebelah kiri dan
f
ketika x
x mendekati % dari sebelah kanan?
.................................... ...................................................... ..................................... ..................................... .................................... ...................................... ................................... ............... .................................... ...................................................... ..................................... ..................................... .................................... ...................................... ................................... ...............
b. "oba "oba tten entu tuka kann nila nilaii fung fungsi si •
f
ketika #
$ilai x mendekati dari sebelah kiri, dengan mengisi tabel dibawah ini. x
3,6
3,7
,'
,(
,( (
,(((
,( ( ( (
,( ( ( ( (
...
→
( x ) ...... f (
......
. . . . ..
......
......
......
......
......
...
→ .. ......
•
$ilai x mendekati dari sebelah kanan, dengan mengisi tabel dibawah ini. x
4,4
4,3
,%
,&
,) &
,) ) &
,) ) ) &
,) ) ) ) &
...
→
( x ) f (
... .......
. . . . . ..
.......
.......
.......
.......
. .. . . . .
.......
→ .. ......
•
+erdasarkan temuan kalian pada tabel, bagaimanakah nilai fungsi mendekati dari sebelah kiri dan
f ketika x
x mendekati dari sebelah kanan?
.................................... ...................................................... ..................................... ..................................... .................................... ...................................... ................................... ............... .................................... ...................................................... ..................................... ..................................... .................................... ...................................... ................................... ............... c. amb ambar arka kanl nlah ah graf grafik ik fun fungs gsii f pada bidang koordinat "artesius dibawah ini.
/i dalam matematika, kata hampir atau mendekati disebut limit. Kegiatan menghitung nilai fungsi disekitar titik dengan mendekati dari sebelah kiri disebut juga −¿
x → a f ( x )
dengan mengitung limit iri dari suatu fungsi dinotasikan
0ntuk kondisi pada soal &1a2, pada soal &1b2,
f ( x ) mempunyai limit ketika
f ( x ) mempunyai limit ketika
f ( x ) ketika x mendekati
lim ¿ ¿
.
x mendekati % dan
x mendekati . 3ehingga, limit
a dapat dinotasikan dengan# lim f ( x ) = L x→ a
%. Perhatikan grafik-grafik pada soal dibawah ini, coba amati dan tentukan limitnya.
Grafk
Grafk
2
a. Pada grafik &, coba tentukan limit kiri dan limit kanan dari
x −1 x f ( x )= x −1 ketika
mendekati &. ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... 2 x −1 x ≠ 1 ( ) lim f x = +erdasarkan penyelidikan kalian, apakah x→ ada? Kemukakan x −1 , 1 alasan kalian4 ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... .......................................................................................................................................................
b. Pada grafik %, coba tentukan limit kiri dan limit kanan dari
{
f ( x )= 3, x ≤ 1 x 1, x > 1 , ketika
mendekati &. ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... 3, x ≤ 1 lim f ( x ) = +erdasarkan penyelidikan kalian, apakah x→ 1 ada? Kemukakan alasan 1, x > 1
{
kalian4 ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... .......................................................................................................................................................
Tahap Pair /iskusikan hasil pengerjaanmu dengan pasanganmu dan coba simpulkanlah hasilnya. 5dakah pendapat kalian yang berbeda? entukan dan tuliskanlah kesepakatan dari perbedaan itu pada ............................................................................................................................ kotak dibawah ini4 ................................. .......................................................................................... ................................................................... ............................................................................................................................ ................................. ............................................................................................................................ ................................. ............................................................................................................................ ................................. ............................................................................................................................
Tahap Share
5pa yang dapat kalian simpulkan dan presentasikan di depan kelas tentang limit fungsi di suatu titik secara umum? ............................................................................................................................ ................................. .......................................................................................... ................................................................... ............................................................................................................................ ................................. ............................................................................................................................ ................................. ............................................................................................................................ ................................. ............................................................................................................................
LEMBAR KEGIATAN SISWA (LKS) 2 Nama: .................................................. Kelas: .................................................. No Absen: ..................................................
A.!. Meng"itung Limit Fungsi Aljabar ujuan Pembelajaran# Peserta didik dapat menghitung limit fungsi aljabar di suatu titik bentuk tak tentu dengan cara-cara penyelesaian yang terdapat pada buku referensi.
Tahap Think 3elesaikanlah soal-soal berikut ini4 "ara apakah yang kalian gunakan untuk menyelesaikan soal-soal berikut ini? 6angan lupa untuk menuliskan alasanmu menggunakan cara tersebut pada kotak jawaban yang tersedia. 2
x − 1 lim &. entukanlah x →−4 x + 1 =… ............................................................................................................................ ................................. .......................................................................................... ................................................................... ............................................................................................................................ ................................. ............................................................................................................................ ................................. ............................................................................................................................ ................................. ............................................................................................................................
%. entukanlah
lim x
2
+ 3 x −4 = …
x→ 2
2
x − 4 lim . entukanlah x →−2 x + 2 =…
2
x − 4 x + 3 =… . entukanlah xlim x −3 →−3
5. entukanlah
lim x → 4
x − 4 =… 1−√ x −3
........................................................................................................................... ........................................................................................................................... .................................................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ...........................................................................................................................
6. entukanlah
lim x→ 2
x −2
√ x + 5 −3 2
=…
........................................................................................................................... ........................................................................................................................... .................................................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ...........................................................................................................................
7. entukanlah
lim x→ 0
(
1
+
1
x x 2− x
)=
…
........................................................................................................................... ........................................................................................................................... .................................................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ...........................................................................................................................
Tahap Pair
/iskusikanlah jawaban kalian dengan pasanganmu. 5dakah pendapat kalian yang berbeda? entukan dan tuliskanlah kesepakatan dari perbedaan itu pada kotak dibawah ini4 ........................................................................................................................... ........................................................................................................................... .................................................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ...........................................................................................................................
Tahap Share +erdasarkan pengerjaan kalian pada soal nomor & sampai dengan nomor 7, simpulkanlah bagaimana cara kalian menentukan cara yang tepat untuk menyelesaikan soal-soal tentang limit fungsi aljabar?
............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ . ........................................................................................................................................................... ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ . ........................................................................................................................................................... ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................
LEMBAR KEGIATAN SISWA (LKS) Nama: .................................................. Kelas: .................................................. No Absen: ..................................................
A.#. Pengertian Limit Fungsi $i Ta %ingga ujuan Pembelajaran# Peserta didik dapat menghitung limit fungsi aljabar di tak hingga.
Tahap Think entukanlah nilai fungsi pada titik berikut ini agar kalian mengetahui pendekatan nilai fungsi ketika x mendekati tak hingga
(∞ ) .
&. 8silah tabel berikut untuk menghitung nilai fungsi x
10
f ( x )
:
f ( x ) = x
2
ketika #
100
&)))
&)9
&)'
&)&9
...
→∞
:
:
...
:
:
:
:
+erdasarkan pengamatan kalian pada tabel, jika nilai nilai
x mendekati tak hingga
( ∞ ) maka
f ( x ) ...................
$otasikan fungsi tersebut ke dalam bentuk limit tak hingga4 ............................................................................................................................................................. ............................................................................................................................................................. f ( x ) =
%. 8silah tabel berikut untuk menghitung nilai fungsi
x ketika #
x
10
100
&).)))
&)9
&)'
&)(
&)&)
→∞
f ( x )
...
:
:
...
:
:
:
:
+erdasarkan pengamatan kalian pada tabel, jika nilai nilai
1
x mendekati tak hingga
( ∞ ) maka
f ( x ) ...................
$otasikan fungsi tersebut ke dalam bentuk limit tak hingga4 ............................................................................................................................................................. ............................................................................................................................................................. x −1 f ( x )= . 8silah tabel berikut untuk menghitung nilai fungsi x −2 ketika # x f ( x )
10
&))
&))))
.....
:
:
&); ......
&)&)
&)&9
&)%)
→∞
:
:
:
→ :
+erdasarkan pengamatan kalian pada tabel, jika nilai nilai
x mendekati tak hingga
( ∞ ) maka
f ( x ) mendekati ...................
$otasikan fungsi tersebut ke dalam bentuk limit tak hingga4 ............................................................................................................................................................. .............................................................................................................................................................
Tahap Pair /iskusikanlah jawabanmu dengan pasanganmu. 5dakah pendapat kalian yang berbeda? entukan dan tuliskanlah kesepakatan dari perbedaan itu pada kotak dibawah ini4
........................................................................................................................... ........................................................................................................................... .................................................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... .................................
Tahap Share
+erdasarkan pengerjaan kalian pada soal nomor & sampai dengan nomor , simpulkanlah bagaimana nilai limit suatu fungsi aljabar ketika
x mendekati tak hingga
(∞ ) ?
........................................................................................................................... ........................................................................................................................... .................................................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ...........................................................................................................................
Tahap Think
Jia
f ( ∞ )=
∞
maa f ( x ) $iuba" $a"ulu $en an &ara $iba i x an at
+erdasarkan konsep yang telah kalian dapatkan dari pengerjaan nomor & sampai dengan nomor , "obalah untuk menentukan penyelesaian dari fungsi-fungsi berikut ini4 .
lim f ( x )= √ x
2
x → ∞
+ 3 x −√ x 2 + x =¿
............
f ( x ) jika kalian mensubstitusikan
a. +erapakah nilai
x = ∞ pada fungsi diatas?
........................................................................................................................... ........................................................................................................................... .................................................................. ........................................................................................................................... ................................. ...........................................................................................................................
b. Perlukah menggunakan cara lain untuk mengetahui nilai limit dari fungsi tersebut? "ara apakah yang dapat kalian gunakan? ........................................................................................................................... ........................................................................................................................... .................................................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. 3
9.
lim x → ∞
2
2 x + 2 x + 4 x
x
3
2
− 2 x + 3 x
a. +erapakah nilai
=¿
.............
f ( x ) jika kalian mensubstitusikan x =∞ pada fungsi diatas?
........................................................................................................................... ........................................................................................................................... .................................................................. ........................................................................................................................... ................................. ...........................................................................................................................
b. Perlukah menggunakan cara lain untuk mengetahui nilai limit dari fungsi tersebut? "ara apakah yang dapat kalian gunakan?
........................................................................................................................... ........................................................................................................................... .................................................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... .................................
Tahap Pair /iskusikanlah jawabanmu dengan pasanganmu. 5dakah pendapat kalian yang berbeda? entukan dan tuliskanlah kesepakatan dari perbedaan itu pada kotak dibawah ini4 ........................................................................................................................... ........................................................................................................................... .................................................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ...........................................................................................................................
Tahap Share
+erdasarkan pengerjaan kalian pada soal nomor sampai dengan nomor 9, simpulkanlah bagaimana langkah-langkah kalian untuk mencari nilai limit dari suatu fungsi ketika x mendekati ∞ ? ........................................................................................................................... ........................................................................................................................... .................................................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ...........................................................................................................................
LATI!AN KEL"M#"K MATERI LIMIT $%NGSI AL&ABAR Pertemuan ke-1 Nama Anggota Kelompok : 1. '''''''''''''''''''''''. ''''''''''''''''''''''' 2. '''''''''''''''''''''''5. ''''''''''''''''''''''' . '''''''''''''''''''''''6. '''''''''''''''''''''''
(isusianla" s)al*s)al beriut ini bersama teman seel)m')mu+ emu$ian 'resentasian "asil erjamu, &2 "ermati grafik fungsi berikut dan apakah fungsi
f mempunyai limit ketika x
2 /iketahui
f ( x )=¿
{
2
x + 2 x ; x≠ 0 x 5 ; x =0
a. entukan nilai fungsi di titik )4 b. 5pakah fungsi diatas mempunyai nilai limit ketika x mendekati )? Kemukakan alasanmu4 2 entukanlah apakah mempunyai limit ketika mendekati &? Kemukakan alasanmu4 %2
−3 92 ;2
2 f ( x )= x + 4 x −2
x mendekati
? Kemukakan alasanmu4
72
Ja-ab : ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... '2 (2
&)2 &&2 &%2
LATI!AN KEL"M#"K MATERI LIMIT $%NGSI AL&ABAR Pertemuan ke-2 Nama Anggota Kelompok : 1. '''''''''''''''''''''''. ''''''''''''''''''''''' 2. '''''''''''''''''''''''5. ''''''''''''''''''''''' . '''''''''''''''''''''''6. '''''''''''''''''''''''
&2 1/ (isusianla" s)al*s)al beriut ini bersama teman seel)m')mu+ emu$ian 'resentasian "asil erjamu. Jangan lu'a untu menulisan se&ara lenga' &ara 'en0elesaiann0a, 1/ 2 x −25 =… lim &. x→ 5 x − 5 %.
.
lim x→ 3
lim x→ 5
2− √ x + 1 =… x −3
x − 5 =… 2 x + 1
.
lim x→ 2
4 2
x −4
−
1
x −2
=…
&;2
Ja-ab : ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ...........................................................................................................................
17) T%GAS R%MA! 1 12/ Nama : .................................................................. 13/ N).Absen : .................................
%)2
&. "ermati grafik fungsi berikut dan tentukan f ( x ) =¿ lim ¿ x→ 3
:.
!1/ Ja-ab: !!/
%2
%. "ermati grafik fungsi berikut dan tentukan f ( x ) =¿ lim ¿
:
x→ 2
!/ Ja-ab:
%92
. "ermati grafik fungsi berikut dan tentukan f ( x ) =¿ lim ¿
:
x→ 0
!4/ Ja-ab:
27) 2) 2*) +) T%GAS R%MA! 2
#1/ Nama : .................................................................. #!/ N).Absen : ................................. 2
#/ Selesaian s)al*s)al beriut ini $engan benar. Tulisan &ara alian mengerjaan $engan lenga', 2 9. x −3 x = lim … &. x→ 3 x −3 ;. x − 5 ( 3 x −1 )2−4 = lim … =… lim 2 %. 7. x→ 5 2 x + 1 x→ 1 x + 4 x − 5 . .
3− √ 9−9 x =… lim 3 x x→ 0
'.
lim x→ 2
4
x
2
−4
−
1
x −2
=…
*.
Ja-ab : ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ........................................................................................................................... ................................. ......................................................................................... ................................................................... ....................................................... .....................................................................................................
1+. TES S"AL LIMIT $%NGSI AL&ABAR 11. Wa,- : 5 men/12. 1#. 5erjaan s)al*s)al $i ba-a" ini $engan benar,
3
&. entukanlah apa fungsi
x − 1 f ( x ) = x −1 yang digambarkan grafik di bawah ini mempunyai limit
ketika x → 1 ? 6elaskanlah jawabanmu4
1. 2
x −9 lim , berikan alasanmu? x→ 3 x − 3
%. unjukkan cara yang kamu gunakan untuk menyelesaikan &9.
14. Untu s)al n). # 6 tentuan benar 7B/ atau sala" 7S/ 'ern0ataan*'ern0ataan beriut ini. Tulisanla" alasan ja-abanmu, 9 − x
2
4 −√ x + 7 lim ¿ 2
. 7B8S/ *elalui cara substitusi diperoleh
=¿ 8
x→ 3
. 7B8S/ /engan cara pemfaktoran diperoleh
lim x→ 2
9. 7B8S/ *elalui cara penyederhanaan diperoleh
(
6 − x 2
x − 4
−
1
x −2
( x + 3 )2− 25 56 =¿ 12 x 2 + 2 x −3 lim ¿ x →3
17.
)=
2