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Pertanyaan

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as y rightwards arrow negative x of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 2 x squared plus 5 x y plus 3 y squared over denominator 3 x squared plus x y minus 2 y squared end fraction end cell end table equals... 

Pembahasan Soal:

Untuk menetukan limit di atas, kita daoat memfaktorkan terlebih dahulu:

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

Jadi, nilai limit tersebut adala negative 1 fifth

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

M. Nasrullah

Mahasiswa/Alumni Universitas Negeri Makassar

Terakhir diupdate 06 Juni 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Dengan menggunakan strategi, tentukan nilai limit fungsi berikut:

Pembahasan Soal:

Limit di atas dapat diselesaikan dengan mensubstitusikan x equals 1 ke dalam fungsi.

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 1 of space fraction numerator x squared plus 2 x minus 3 over denominator 2 x minus 3 end fraction end cell row blank equals cell fraction numerator 1 squared plus 2 times 1 minus 3 over denominator 2 times 1 minus 3 end fraction end cell row blank equals cell fraction numerator 1 plus 2 minus 3 over denominator 2 minus 3 end fraction end cell row blank equals cell fraction numerator 0 over denominator negative 1 end fraction end cell row blank equals 0 end table 

Jadi, hasil dari limit tersebut adalah 0.

0

Roboguru

Jika , maka

Pembahasan Soal:

Diketahui:

begin mathsize 14px style limit as x rightwards arrow a of fraction numerator f open parentheses x close parentheses minus f open parentheses a close parentheses over denominator x minus a end fraction equals 1 end style

Gunakan dalil L'Hospital sehingga diperoleh:

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

Akan ditentukan nilai begin mathsize 14px style limit as x rightwards arrow a of fraction numerator a f open parentheses x close parentheses minus x f open parentheses a close parentheses over denominator x minus a end fraction end style dengan menggunakan dalil L'Hospital, diperoleh sebagai berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell limit as x rightwards arrow a of fraction numerator a f open parentheses x close parentheses minus x f open parentheses a close parentheses over denominator x minus a end fraction end cell equals cell limit as x rightwards arrow a of fraction numerator a f apostrophe open parentheses x close parentheses minus 1 times f open parentheses a close parentheses over denominator 1 minus 0 end fraction end cell row blank equals cell limit as x rightwards arrow a of fraction numerator a f apostrophe open parentheses x close parentheses minus f open parentheses a close parentheses over denominator 1 end fraction end cell row blank equals cell a f apostrophe open parentheses a close parentheses minus f open parentheses a close parentheses space space space open parentheses substitusi space nilai space f apostrophe open parentheses a close parentheses equals 1 close parentheses end cell row blank equals cell a times 1 minus f open parentheses a close parentheses end cell row cell limit as x rightwards arrow a of fraction numerator a f open parentheses x close parentheses minus x f open parentheses a close parentheses over denominator x minus a end fraction end cell equals cell a minus f open parentheses a close parentheses end cell end table end style 

Jadi, jawaban yang tepat adalah E.

0

Roboguru

Pembahasan Soal:

Diketahui:

Limit bentuk akar pecahan.

Cara pertama untuk mengetahui suatu nilai limit fungsi adalah dengan melakukan substitusi nilai limit begin mathsize 14px style x end style pada persamaan. Jika hasil yang diperoleh adalah suatu nilai, (bukan begin mathsize 14px style 0 over 0 end style) maka proses mencari nilai limit selesai.

Substitusi variabel begin mathsize 14px style x end style pada limit dengan begin mathsize 14px style 2 end style seperti berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 2 of space square root of fraction numerator x cubed plus 2 x plus 3 over denominator x squared plus 5 end fraction end root end cell equals cell space square root of fraction numerator 2 cubed plus 2 open parentheses 2 close parentheses plus 3 over denominator 2 squared plus 5 end fraction end root end cell row blank equals cell square root of fraction numerator 8 plus 4 plus 3 over denominator 4 plus 9 end fraction end root end cell row blank equals cell square root of 15 over 9 end root end cell row blank equals cell square root of 15 over 3 squared end root end cell row blank equals cell fraction numerator square root of 15 over denominator 3 end fraction end cell end table end style

Maka, begin mathsize 14px style limit as x rightwards arrow 2 of space square root of fraction numerator x cubed plus 2 x plus 3 over denominator x squared plus 5 end fraction end root equals fraction numerator square root of 15 over denominator 3 end fraction end style.

0

Roboguru

Pembahasan Soal:

Diketahui:

Limit bentuk akar pecahan.

Cara pertama untuk mengetahui suatu nilai limit fungsi adalah dengan melakukan substitusi nilai limit begin mathsize 14px style x end style pada persamaan. Jika hasil yang diperoleh adalah suatu nilai, (bukan begin mathsize 14px style 0 over 0 end style) maka proses selesai.

Substitusi variabel begin mathsize 14px style x end style pada limit dengan begin mathsize 14px style 2 end style seperti berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 2 of space square root of fraction numerator x squared plus 3 x plus 4 over denominator x cubed plus 1 end fraction end root end cell equals cell space square root of fraction numerator 2 squared plus 3 open parentheses 2 close parentheses plus 4 over denominator 2 cubed plus 1 end fraction end root end cell row blank equals cell square root of fraction numerator 4 plus 6 plus 4 over denominator 8 plus 1 end fraction end root end cell row blank equals cell square root of 14 over 9 end root end cell row blank equals cell square root of 14 over 3 squared end root end cell row blank equals cell fraction numerator square root of 14 over denominator 3 end fraction end cell end table end style

Maka, begin mathsize 14px style limit as x rightwards arrow 2 of space square root of fraction numerator x squared plus 3 x plus 4 over denominator x cubed plus 1 end fraction end root equals fraction numerator square root of 14 over denominator 3 end fraction end style.

0

Roboguru

Jika , 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 a x plus b minus square root of x end style dan begin mathsize 14px style x minus 4 end style.

Turunan fungsi begin mathsize 14px style a x plus b minus square root of x end style:

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

Turunan fungsi begin mathsize 14px style x minus 4 end style:

begin mathsize 14px style y equals x minus 4 y apostrophe equals 1 end style    

Tentukan nilai begin mathsize 14px style a end style:

  begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 4 of fraction numerator a x plus b minus square root of x over denominator x minus 4 end fraction end cell equals cell 3 over 4 end cell row cell limit as x rightwards arrow 4 of fraction numerator a minus begin display style fraction numerator 1 over denominator 2 square root of x end fraction end style over denominator 1 end fraction end cell equals cell 3 over 4 end cell row cell limit as x rightwards arrow 4 of space a minus fraction numerator 1 over denominator 2 square root of x end fraction end cell equals cell 3 over 4 end cell row cell a minus fraction numerator 1 over denominator 2 square root of 4 end fraction end cell equals cell 3 over 4 end cell row cell a minus 1 fourth end cell equals cell 3 over 4 end cell row a equals cell 3 over 4 plus 1 fourth end cell row a equals 1 end table end style  

Substitusikan nilai begin mathsize 14px style a equals 1 end style ke begin mathsize 14px style limit as x rightwards arrow 4 of fraction numerator a x plus b minus square root of x over denominator x minus 4 end fraction equals 3 over 4 end style.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 4 of fraction numerator a x plus b minus square root of x over denominator x minus 4 end fraction end cell equals cell 3 over 4 end cell row cell limit as x rightwards arrow 4 of fraction numerator x plus b minus square root of x over denominator x minus 4 end fraction end cell equals cell 3 over 4 end cell row cell fraction numerator 4 plus b minus square root of 4 over denominator 4 minus 4 end fraction end cell equals cell 3 over 4 end cell row cell 4 open parentheses 4 plus b minus square root of 4 close parentheses end cell equals cell 3 left parenthesis 4 minus 4 right parenthesis end cell row cell 4 left parenthesis 4 plus b minus 2 right parenthesis end cell equals 0 row cell 16 plus 4 b minus 8 end cell equals 0 row cell 4 b plus 8 end cell equals 0 row cell 4 b end cell equals cell negative 8 end cell row b equals cell negative 8 over 4 end cell row b equals cell negative 2 end cell end table end style 

Nilai begin mathsize 14px style a plus b end style:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell a plus b end cell equals cell 1 minus 2 end cell row blank equals cell negative 1 end cell end table end style 

Sehingga, jika begin mathsize 14px style limit as x rightwards arrow 4 of fraction numerator a x plus b minus square root of x over denominator x minus 4 end fraction equals 3 over 4 end style maka nilai begin mathsize 14px style a plus b end style adalah begin mathsize 14px style negative 1 end style.

Jadi, jawaban yang tepat adalah B.

0

Roboguru

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