Roboguru

Pertanyaan

limit as x rightwards arrow 2 of space fraction numerator x cubed minus 8 over denominator x squared plus x minus 6 end fraction equals space...

Pembahasan Soal:

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 2 of space fraction numerator x cubed minus 8 over denominator x squared plus x minus 6 end fraction end cell equals cell space limit as x rightwards arrow 2 of space fraction numerator left parenthesis x minus 2 right parenthesis left parenthesis x squared plus 2 x plus 4 right parenthesis over denominator left parenthesis x minus 2 right parenthesis left parenthesis x plus 3 right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 2 of space fraction numerator left parenthesis x squared plus 2 x plus 4 right parenthesis over denominator left parenthesis x plus 3 right parenthesis end fraction end cell row blank equals cell fraction numerator left parenthesis 2 right parenthesis squared plus 2 left parenthesis 2 right parenthesis plus 4 over denominator 2 plus 3 end fraction end cell row blank equals cell 12 over 5 end cell end table

Dengan demikian, hasil dari limit as x rightwards arrow 2 of space fraction numerator x cubed minus 8 over denominator x squared plus x minus 6 end fraction equals space 12 over 5.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

N. Indah

Mahasiswa/Alumni Universitas Diponegoro

Terakhir diupdate 05 Juni 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

Tapi Roboguru masih mau belajar. Menurut kamu pembahasan kali ini sudah membantu, belum?

Membantu

Kurang Membantu

Apakah pembahasan ini membantu?

Belum menemukan yang kamu cari?

Post pertanyaanmu ke Tanya Jawab, yuk

Mau Bertanya

Pertanyaan yang serupa

Nilai  adalah ....

Pembahasan Soal:

Limit merupakan sebuah konsep matematika dimana sesuatu dikatakan "hampir" atau "mendekati" nilai suatu bilangan tertentu.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 4 of space fraction numerator 6 x square root of x minus 24 square root of x over denominator square root of x minus 2 end fraction end cell equals cell limit as x rightwards arrow 4 of space fraction numerator 6 square root of x open parentheses x minus 4 close parentheses over denominator square root of x minus 2 end fraction end cell row blank equals cell limit as x rightwards arrow 4 of space fraction numerator 6 square root of x left parenthesis x minus 4 right parenthesis over denominator square root of x minus 2 end fraction cross times fraction numerator square root of x plus 2 over denominator square root of x plus 2 end fraction end cell row blank equals cell limit as x rightwards arrow 4 of space fraction numerator 6 square root of x up diagonal strike left parenthesis x minus 4 right parenthesis end strike open parentheses square root of x plus 2 close parentheses over denominator up diagonal strike x minus 4 end strike end fraction end cell row blank equals cell limit as x rightwards arrow 4 of space 6 square root of x open parentheses square root of x plus 2 close parentheses end cell row blank equals cell 6 square root of 4 open parentheses square root of 4 plus 2 close parentheses end cell row blank equals cell 6 times 2 open parentheses 2 plus 2 close parentheses end cell row blank equals cell 12 times 4 end cell row blank equals 48 end table end style 

Jadi, nilai dari begin mathsize 14px style limit as x rightwards arrow 4 of space fraction numerator 6 x square root of x minus 24 square root of x over denominator square root of x minus 2 end fraction end style adalah begin mathsize 14px style 48 end style

Roboguru

Nilai  adalah ...

Pembahasan Soal:

Limit merupakan sebuah konsep matematika dimana sesuatu dikatakan "hampir" atau "mendekati" nilai suatu bilangan tertentu.

 Dengan menggunakan dalil L'Hopital, dimana:

begin mathsize 14px style limit as x rightwards arrow a of fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction equals limit as x rightwards arrow a of fraction numerator f apostrophe left parenthesis x right parenthesis over denominator g apostrophe left parenthesis x right parenthesis end fraction end style

maka tentukan turunan dari begin mathsize 14px style 2 x squared minus 162 end style dan begin mathsize 14px style x plus 3 square root of x minus 18 end style.

Turunan fungsi begin mathsize 14px style 2 x squared minus 162 end style:

begin mathsize 14px style y equals 2 x squared minus 162 y apostrophe equals 4 x end style 

Turunan fungsi begin mathsize 14px style x plus 3 square root of x minus 18 end style:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell x plus 3 square root of x minus 18 end cell row y equals cell x plus 3 x to the power of begin inline style 1 half end style end exponent minus 18 end cell row cell y apostrophe end cell equals cell 1 plus 3 times 1 half x to the power of begin inline style 1 half end style minus 1 end exponent end cell row cell y apostrophe end cell equals cell 1 plus 3 over 2 x to the power of negative begin inline style 1 half end style end exponent end cell row cell y apostrophe end cell equals cell 1 plus 3 fraction numerator 3 over denominator 2 square root of x end fraction end cell end table end style  

Nilai begin mathsize 14px style limit as x rightwards arrow 9 of space fraction numerator 2 x squared minus 162 over denominator x plus 3 square root of x minus 18 end fraction end style:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 9 of space fraction numerator 2 x squared minus 162 over denominator x plus 3 square root of x minus 18 end fraction end cell equals cell limit as x rightwards arrow 9 of space fraction numerator 4 x over denominator 1 plus begin display style fraction numerator 3 over denominator 2 square root of x end fraction end style end fraction end cell row blank equals cell fraction numerator 4 left parenthesis 9 right parenthesis over denominator 1 plus begin display style fraction numerator 3 over denominator 2 square root of 9 end fraction end style end fraction end cell row blank equals cell fraction numerator 36 over denominator 1 plus begin display style fraction numerator 3 over denominator 2 times 3 end fraction end style end fraction end cell row blank equals cell fraction numerator 36 over denominator 1 plus begin display style 3 over 6 end style end fraction end cell row blank equals cell fraction numerator 36 over denominator begin display style 9 over 6 end style end fraction end cell row blank equals cell fraction numerator 36 over denominator begin display style 3 over 2 end style end fraction end cell row blank equals cell 36 cross times 2 over 3 end cell row blank equals 24 end table end style  

Sehingga, nilai dari begin mathsize 14px style limit as x rightwards arrow 9 of space fraction numerator 2 x squared minus 162 over denominator x plus 3 square root of x minus 18 end fraction end style adalah begin mathsize 14px style 24 end style

Jadi, jawaban yang tepat adalah B.

Roboguru

...

Pembahasan Soal:

Gunakan pemfaktoran untuk menyederhanakan fungsi limit.

begin mathsize 14px style table attributes columnalign center center left end attributes row cell limit as x rightwards arrow negative 7 of space fraction numerator x squared plus 5 x minus 14 over denominator x squared plus 11 x plus 28 end fraction end cell equals cell limit as x rightwards arrow negative 7 of space fraction numerator x squared plus 7 x minus 2 x minus 14 over denominator x squared plus 7 x plus 4 x plus 28 end fraction end cell row space equals cell limit as x rightwards arrow negative 7 of space fraction numerator x left parenthesis x plus 7 right parenthesis minus 2 left parenthesis x plus 7 right parenthesis over denominator x left parenthesis x plus 7 right parenthesis plus 4 left parenthesis x plus 7 right parenthesis end fraction end cell row space equals cell limit as x rightwards arrow negative 7 of space fraction numerator down diagonal strike left parenthesis x plus 7 right parenthesis end strike left parenthesis x minus 2 right parenthesis over denominator down diagonal strike left parenthesis x plus 7 right parenthesis end strike left parenthesis x plus 4 right parenthesis end fraction end cell row space equals cell limit as x rightwards arrow negative 7 of space fraction numerator x minus 2 over denominator x plus 4 end fraction end cell row space space space row space space space row space space cell space space end cell end table end style 

Kemudian substitusi begin mathsize 14px style x equals negative 7 end style ke seperti berikut:

begin mathsize 14px style table attributes columnalign center center left end attributes row cell limit as x rightwards arrow negative 7 of space fraction numerator x minus 2 over denominator x plus 4 end fraction end cell equals cell fraction numerator open parentheses negative 7 close parentheses minus 2 over denominator open parentheses negative 7 close parentheses plus 4 end fraction end cell row space equals cell fraction numerator negative 9 over denominator negative 3 end fraction end cell row space equals 3 end table end style 

 

Roboguru

...

Pembahasan Soal:

Untuk menentukan nilai limit, jika menggunakan metode substitusi untuk nilai begin mathsize 14px style x equals 3 end style maka diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x left right arrow 3 of fraction numerator x squared minus 9 over denominator x squared minus x minus 6 end fraction end cell equals cell fraction numerator 3 squared minus 9 over denominator 3 squared minus 3 minus 6 end fraction end cell row blank equals cell fraction numerator 9 minus 9 over denominator 9 minus 3 minus 6 end fraction end cell row blank equals cell 0 over 0 end cell end table end style 

Oleh karena menggunakan metode substitusi diperoleh begin mathsize 14px style 0 over 0 end style, maka untuk mendapat nilai limit begin mathsize 14px style x end style mendekati 3 dapat menggunakan metode pemfaktoran seperti berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 3 of fraction numerator x squared minus 9 over denominator x squared minus x minus 6 end fraction end cell equals cell limit as x rightwards arrow 3 of fraction numerator open parentheses x minus 3 close parentheses open parentheses x plus 3 close parentheses over denominator open parentheses x minus 3 close parentheses open parentheses x plus 2 close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 3 of fraction numerator x plus 3 over denominator x plus 2 end fraction end cell row blank equals cell fraction numerator 3 plus 3 over denominator 2 plus 2 end fraction end cell row blank equals cell 6 over 5 end cell end table end style 

Dengan demikian, nilai begin mathsize 14px style limit as x rightwards arrow 3 of fraction numerator x squared minus 9 over denominator x squared minus x minus 9 end fraction end style adalah begin mathsize 14px style 6 over 5 end style.

Oleh karena itu, jawaban yang benar adalah D.

Roboguru

Hitunglah nilai limit berikut ini.

Pembahasan Soal:

Nilai dari begin mathsize 14px style limit as x rightwards arrow 2 of fraction numerator 2 x squared minus 5 x plus 2 over denominator x minus 2 end fraction end style dapat ditentukan dengan menggunakan metode pemfaktoran.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 2 of fraction numerator 2 x squared minus 5 x plus 2 over denominator x minus 2 end fraction end cell equals cell limit as x rightwards arrow 2 of fraction numerator open parentheses 2 x minus 1 close parentheses open parentheses x minus 2 close parentheses over denominator x minus 2 end fraction end cell row blank equals cell limit as x rightwards arrow 2 of open parentheses 2 x minus 1 close parentheses end cell row blank equals cell 2 open parentheses 2 close parentheses minus 1 end cell row blank equals 3 end table end style

Dengan demikian, nilai begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 2 of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 2 x squared minus 5 x plus 2 over denominator x minus 2 end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table end style

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

Tapi Roboguru masih mau belajar. Menurut kamu pembahasan kali ini sudah membantu, belum?

Membantu

Kurang Membantu

Apakah pembahasan ini membantu?

Belum menemukan yang kamu cari?

Post pertanyaanmu ke Tanya Jawab, yuk

Mau Bertanya

RUANGGURU HQ

Jl. Dr. Saharjo No.161, Manggarai Selatan, Tebet, Kota Jakarta Selatan, Daerah Khusus Ibukota Jakarta 12860

Coba GRATIS Aplikasi Ruangguru

Produk Ruangguru

Produk Lainnya

Hubungi Kami

Ikuti Kami

©2021 Ruangguru. All Rights Reserved