LKS Limit SMA Kelas 1

LEMBAR KEGIATAN SISWA (LKS) 1

A. LIMIT FUNGSI ALJABARKalkulus 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 kalkulusPada lembar kerja siswa berikut ini didesain kegiatan siswa yang melalui tahapan Think --> Pair Share. Tujuan 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 mengevaluasi.

A.1. Pengertian Limit Fungsi Aljabar

1. Cermatilah fungsi berikut ini f ( x )=x+1a. Coba tentukan nilai fungsi f ketika :

Nilai x mendekati 2 dari sebelah kiri, dengan mengisi tabel dibawah ini.

x 1 ,6 1,7 1,8 1,9 1,99 1,999 1,9999 1,99999 ... → 2f (x) ...... ...... ...... ...... ...... ...... ...... ...... ... →........

Nilai x mendekati 2 dari sebelah kanan, dengan mengisi tabel dibawah ini.

x 2,4 2,3 2,2 2,1 2,01 2,001 2,0001 2,00001 ... → 2f (x) ...... ...... ...... ...... ...... ...... ...... ...... ... →........

Berdasarkan temuan kalian pada tabel, bagaimanakah nilai fungsi f ketika x mendekati 2 dari sebelah kiri dan x mendekati 2 dari sebelah kanan?................................................................................................................................................................................................................................................................................................

Tujuan Pembelajaran: Peserta didik dapat memahami dan menghitung limit fungsi aljabar di suatu titik secara intuitif.

Nama: ..................................................

Kelas: ..................................................

No Absen

Tahap Think

Untuk kondisi pada soal 1(a), f (x) mempunyai limit ketika x mendekati 2 dan pada soal 1(b), f ( x ) mempunyai limit ketika x mendekati 4. Sehingga, limit f (x) ketika x mendekati a dapat dinotasikan dengan:

limx→a

f ( x )=L

Coba simpulkan, suatu fungsi f akan mempunyai limit jika:...........................................................................................................................................................................................................................................................................................

b. Coba tentukan nilai fungsi f ketika : Nilai x mendekati 4 dari sebelah kiri, dengan mengisi tabel dibawah ini.

x 3 ,6 3,7 3,8 3,9 3,99 3,999 3,9999 3,99999 ... → 4f (x) ...... ...... ...... ...... ...... ...... ...... ...... ... →........

Nilai x mendekati 4 dari sebelah kanan, dengan mengisi tabel dibawah ini.

x 4,4 4,3 4,2 4,1 4,01 4,001 4,0001 4,00001 ... → 4f (x) ....... ....... ....... ....... ....... ....... ....... ....... ... →........

Berdasarkan temuan kalian pada tabel, bagaimanakah nilai fungsi f ketika x mendekati 4 dari sebelah kiri dan x mendekati 4 dari sebelah kanan?................................................................................................................................................................................................................................................................................................

c. Gambarkanlah grafik fungsi f pada bidang koordinat Cartesius dibawah ini.

Di dalam matematika, kata hampir atau mendekati disebut limit. Kegiatan menghitung nilai fungsi disekitar titik dengan mendekati dari sebelah kiri disebut juga

dengan mengitung limit kiri dari suatu fungsi dinotasikan limx→a−¿ f (x)¿

¿.

Kegiatan menghitung nilai fungsi disekitar titik dengan mendekati dari sebelah kanan disebut

juga dengan mengitung limit kanan dari suatu fungsi dinotasikan limx→a+¿ f (x)¿

¿.

2. Perhatikan grafik-grafik pada soal dibawah ini, coba amati dan tentukan limitnya.

a. Pada grafik 1, coba tentukan limit kiri dan limit kanan dari f ( x )= x2−1x−1

ketika x mendekati 1.

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Berdasarkan penyelidikan kalian, apakah limx→1

f ( x )= x2−1x−1

, x≠1 ada? Kemukakan alasan

kalian!............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

b. Pada grafik 2, coba tentukan limit kiri dan limit kanan dari f ( x )={3 , x≤11 , x>1 , ketika x mendekati

1...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Berdasarkan penyelidikan kalian, apakah limx→1

f ( x )={3 , x ≤11 , x>1 ada? Kemukakan alasan

kalian!............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Grafik 1 Grafik 2

Diskusikan hasil pengerjaanmu dengan pasanganmu dan coba simpulkanlah hasilnya. Adakah pendapat kalian yang berbeda? Tentukan dan tuliskanlah kesepakatan dari perbedaan itu pada kotak dibawah ini!

Apa yang dapat kalian simpulkan dan presentasikan di depan kelas tentang limit fungsi di suatu titik secara umum?

LEMBAR KEGIATAN SISWA (LKS) 2

Tahap Pair

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Tahap Share

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Nama: ..................................................

Kelas: ..................................................

No Absen

A.2. Menghitung Limit Fungsi Aljabar

Selesaikanlah soal-soal berikut ini! Cara apakah yang kalian gunakan untuk menyelesaikan soal-soal berikut ini? Jangan lupa untuk menuliskan alasanmu menggunakan cara tersebut pada kotak jawaban yang tersedia.

1. Tentukanlah limx→−4

x2−1x+1

=…

2. Tentukanlah limx→2

x2+3 x−4=…

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3. Tentukanlah limx→−2

x2−4x+2

=…

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Tujuan 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

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4. Tentukanlah limx→−3

x2−4 x+3x−3

=…

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5. Tentukanlah limx→ 4

x−41−√x−3

=…

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6. Tentukanlah limx→2

x−2

√ x2+5−3=…

7. Tentukanlah limx→0 ( 1x + 1

x2−x )=…

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Diskusikanlah jawaban kalian dengan pasanganmu. Adakah pendapat kalian yang berbeda? Tentukan dan tuliskanlah kesepakatan dari perbedaan itu pada kotak dibawah ini!

Berdasarkan pengerjaan kalian pada soal nomor 1 sampai dengan nomor 7, simpulkanlah bagaimana cara kalian menentukan cara yang tepat untuk menyelesaikan soal-soal tentang limit fungsi aljabar?

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Tahap Pair

Tahap Share

LEMBAR KEGIATAN SISWA (LKS) 3

A.3. Pengertian Limit Fungsi di Tak Hingga

Tentukanlah nilai fungsi pada titik berikut ini agar kalian mengetahui pendekatan nilai fungsi ketika x mendekati tak hingga (∞).

1. Isilah tabel berikut untuk menghitung nilai fungsi f ( x )=x2 ketika :

x 10 100 1000 105 108 1015 ... →∞f (x) … … … ... … … … …

Berdasarkan pengamatan kalian pada tabel, jika nilai x mendekati tak hingga (∞) maka nilai f (x)...................Notasikan fungsi tersebut ke dalam bentuk limit tak hingga!..........................................................................................................................................................................................................................................................................................................................

2. Isilah tabel berikut untuk menghitung nilai fungsi f ( x )=1x

ketika :

x 10 100 10.000 105 108 109 1010 →∞f (x) ... … … ... … … … …

Berdasarkan pengamatan kalian pada tabel, jika nilai x mendekati tak hingga (∞) maka nilai f (x)...................Notasikan fungsi tersebut ke dalam bentuk limit tak hingga!..........................................................................................................................................................................................................................................................................................................................

3. Isilah tabel berikut untuk menghitung nilai fungsi f ( x )= x−1x−2

ketika :

x 10 100 10000 106 1010 1015 1020 →∞f (x) ..... … … ...... … … … → …

Berdasarkan pengamatan kalian pada tabel, jika nilai x mendekati tak hingga (∞) maka nilai f (x) mendekati ...................Notasikan fungsi tersebut ke dalam bentuk limit tak hingga!..........................................................................................................................................................................................................................................................................................................................

Nama: ..................................................

Kelas: ..................................................

No Absen

Tujuan Pembelajaran: Peserta didik dapat menghitung limit fungsi aljabar di tak hingga.

Tahap Think

Diskusikanlah jawabanmu dengan pasanganmu. Adakah pendapat kalian yang berbeda? Tentukan dan tuliskanlah kesepakatan dari perbedaan itu pada kotak dibawah ini!

Berdasarkan pengerjaan kalian pada soal nomor 1 sampai dengan nomor 3, simpulkanlah bagaimana nilai limit suatu fungsi aljabar ketika x mendekati tak hingga (∞)?

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Tahap Pair

Tahap Share

Berdasarkan konsep yang telah kalian dapatkan dari pengerjaan nomor 1 sampai dengan nomor 3, Cobalah untuk menentukan penyelesaian dari fungsi-fungsi berikut ini!

4. limx→∞

f (x )=√x2+3 x−√ x2+x=¿............

a. Berapakah nilai f (x) jika kalian mensubstitusikan x=∞ pada fungsi diatas?

b. Perlukah menggunakan cara lain untuk mengetahui nilai limit dari fungsi tersebut? Cara apakah yang dapat kalian gunakan?

5. limx→∞

2x3+2 x2+4 xx3−2x2+3 x

=¿.............

a. Berapakah nilai f (x) jika kalian mensubstitusikan x=∞ pada fungsi diatas?

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Tahap Think

Jika f (∞ )=∞∞

maka f (x) diubah dahulu dengan cara dibagi x pangkat terbesar

b. Perlukah menggunakan cara lain untuk mengetahui nilai limit dari fungsi tersebut? Cara apakah yang dapat kalian gunakan?

Diskusikanlah jawabanmu dengan pasanganmu. Adakah pendapat kalian yang berbeda? Tentukan dan tuliskanlah kesepakatan dari perbedaan itu pada kotak dibawah ini!

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Tahap Pair

Berdasarkan pengerjaan kalian pada soal nomor 4 sampai dengan nomor 5, simpulkanlah bagaimana langkah-langkah kalian untuk mencari nilai limit dari suatu fungsi ketika x mendekati ∞?

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Tahap Share

Diskusikanlah soal-soal berikut ini bersama teman sekelompokmu, kemudian presentasikan

hasil kerjamu!

1) Cermati grafik fungsi berikut dan apakah fungsi f mempunyai limit ketika x mendekati 1? Kemukakan alasanmu!

2) Diketahui f ( x )=¿ {x2+2 xx

;x ≠0

5 ; x=0a. Tentukan nilai fungsi di titik 0!b. Apakah fungsi diatas mempunyai nilai

limit ketika x mendekati 0? Kemukakan alasanmu!

3) Tentukanlah apakah f ( x )= x2+4x−2

mempunyai limit ketika x mendekati −3? Kemukakan alasanmu!

LATIHAN KELOMPOKMATERI LIMIT FUNGSI ALJABAR

Pertemuan ke-1Nama Anggota Kelompok :1. _______________________ 4. _______________________2. _______________________ 5. _______________________3. _______________________ 6. _______________________

Diskusikanlah soal-soal berikut ini bersama teman sekelompokmu, kemudian presentasikan hasil kerjamu. Jangan lupa untuk menuliskan secara lengkap cara penyelesaiannya!

1. limx→5

x2−25x−5

=…

2. limx→3

2−√x+1x−3

=…

3. limx→5

x−52 x+1

=…

4. limx→2

4

x2−4− 1x−2

=…

Jawab :........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ ............................................................................................................................................................ ........................................................................................................................................................................................................................................................................................................................ ............................................................................................................................................................

LATIHAN KELOMPOKMATERI LIMIT FUNGSI ALJABAR

Pertemuan ke-2Nama Anggota Kelompok :1. _______________________ 4. _______________________2. _______________________ 5. _______________________3. _______________________ 6. _______________________

TUGAS RUMAH 1

Nama : ..................................................................No.Absen : .................................

1. Cermati grafik fungsi berikut dan tentukan limx→3

f ( x )=¿¿….Jawab:

Jawab :........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ ............................................................................................................................................................ ........................................................................................................................................................................................................................................................................................................................ ............................................................................................................................................................ ................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ ............................................................................................................................................................ ........................................................................................................................................................................................................................................................................................................................ ............................................................................................................................................................

2. Cermati grafik fungsi berikut dan tentukan limx→2

f ( x )=¿¿…

Jawab:

3. Cermati grafik fungsi berikut dan tentukan limx→0

f ( x )=¿¿…

Jawab:

TUGAS RUMAH 2

Nama : ..................................................................No.Absen : .................................

Selesaikan soal-soal berikut ini dengan benar. Tuliskan cara kalian mengerjakan dengan lengkap!

1. limx→3

x2−3 xx−3

=…

2. limx→5

x−52 x+1

=…

3. limx→0

3−√9−9x3 x

=…

4. limx→1

(3 x−1 )2−4x2+4 x−5

=…

5. limx→2

4

x2−4− 1x−2

=…

TES SOAL LIMIT FUNGSI ALJABARWaktu : 45 menit

Kerjakan soal-soal di bawah ini dengan benar!

1. Tentukanlah apa fungsi f ( x )= x3−1x−1

yang digambarkan grafik di bawah ini mempunyai limit

ketika x→1? Jelaskanlah jawabanmu!

Jawab :........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ ............................................................................................................................................................ ........................................................................................................................................................................................................................................................................................................................ ............................................................................................................................................................ ................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ ............................................................................................................................................................ ........................................................................................................................................................................................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ........................................................................................................................................................................................................................................................................................................................ ............................................................................................................................................................

2. Tunjukkan cara yang kamu gunakan untuk menyelesaikan limx→3

x2−9x−3

, berikan alasanmu?

Untuk soal no. 3 – 5 tentukan benar (B) atau salah (S) pernyataan-pernyataan berikut ini. Tuliskanlah alasan jawabanmu!

3. (B/S) Melalui cara substitusi diperoleh limx→3

9−x2

4−√x2+7=¿8¿

4. (B/S) Dengan cara pemfaktoran diperoleh limx→2 ( 6−xx2−4

−1x−2 )=2

5. (B/S) Melalui cara penyederhanaan diperoleh limx→3

( x+3 )2−25x2+2x−3

=¿ 5612

¿