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Jika x → 2 lim ​ x 3 n ​ − 2 3 n ​ x n − 2 n ​ = 6 3 2 ​ , nilai n adalah ...

Jika , nilai  adalah ...

  1. begin mathsize 14px style 1 end style 

  2. begin mathsize 14px style 2 end style 

  3.  begin mathsize 14px style 3 end style  

  4.  begin mathsize 14px style 4 end style 

  5. begin mathsize 14px style 5 end style 

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D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah B.

jawaban yang tepat adalah B.

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Limit merupakan sebuah konsep matematika dimana sesuatu dikatakan "hampir" atau "mendekati" nilai suatu bilangan tertentu. Dengan menggunakan dalil L'Hopital, dimana: maka tentukan turunan dari dan . Turunan fungsi : Turunan fungsi : Maka nilai dapat ditentukan sebagai berikut: Sehingga, jika , nilai adalah . Jadi, jawaban yang tepat adalah B.

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 x to the power of n minus 2 to the power of n end style dan begin mathsize 14px style x to the power of begin inline style n over 3 end style end exponent minus 2 to the power of begin inline style n over 3 end style end exponent end style.

Turunan fungsi begin mathsize 14px style x to the power of n minus 2 to the power of n end style:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell x to the power of n minus 2 to the power of n end cell row cell y apostrophe end cell equals cell n x to the power of n minus 1 end exponent end cell end table end style      

Turunan fungsi begin mathsize 14px style x to the power of begin inline style n over 3 end style end exponent minus 2 to the power of begin inline style n over 3 end style end exponent end style:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell x to the power of begin inline style n over 3 end style end exponent minus 2 to the power of begin inline style n over 3 end style end exponent end cell row cell y apostrophe end cell equals cell n over 3 x to the power of begin inline style n over 3 end style minus 1 end exponent end cell end table end style     

 Maka nilai begin mathsize 14px style n end style dapat ditentukan sebagai berikut:

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 x to the power of n minus 2 to the power of n over denominator x to the power of n over 3 end exponent minus 2 to the power of n over 3 end exponent end fraction end cell equals cell 6 cube root of 2 end cell row cell limit as x rightwards arrow 2 of fraction numerator up diagonal strike n x to the power of n minus 1 end exponent over denominator begin display style fraction numerator up diagonal strike n over denominator 3 end fraction end style x to the power of n over 3 minus 1 end exponent end fraction end cell equals cell 6 cube root of 2 end cell row cell limit as x rightwards arrow 2 of fraction numerator x to the power of n minus 1 end exponent over denominator begin display style 1 third x to the power of n over 3 minus 1 end exponent end style end fraction end cell equals cell 6 cube root of 2 end cell row cell limit as x rightwards arrow 2 of space 3 x to the power of n minus 1 minus n over 3 plus 1 end exponent end cell equals cell 6 cube root of 2 end cell row cell limit as x rightwards arrow 2 of space 3 x to the power of n minus n over 3 end exponent end cell equals cell 6 cube root of 2 end cell row cell 3 times 2 to the power of n minus n over 3 end exponent end cell equals cell 6 cube root of 2 end cell row cell up diagonal strike 3 times 2 to the power of fraction numerator 2 n over denominator 3 end fraction end exponent end cell equals cell up diagonal strike 3 times 2 times 2 to the power of begin inline style 1 third end style end exponent end cell row cell 2 to the power of fraction numerator 2 n over denominator 3 end fraction end exponent end cell equals cell 2 times 2 to the power of begin inline style 1 third end style end exponent end cell row cell 2 to the power of fraction numerator 2 n over denominator 3 end fraction end exponent end cell equals cell 2 to the power of begin inline style 4 over 3 end style end exponent end cell row cell 2 n end cell equals 4 row n equals 2 end table end style 

Sehingga, jika begin mathsize 14px style limit as x rightwards arrow 2 of fraction numerator x to the power of n minus 2 to the power of n over denominator x to the power of n over 3 end exponent minus 2 to the power of n over 3 end exponent end fraction equals 6 cube root of 2 end style, nilai begin mathsize 14px style n end style adalah begin mathsize 14px style 2 end style

Jadi, jawaban yang tepat adalah B.

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