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Tentukan nilai x→∞lim​3x+2 sin x−8cos 4x− 6x​!

Pertanyaan

Tentukan nilai limit as x rightwards arrow infinity of fraction numerator cos space 4 x minus space 6 x over denominator 3 x plus 2 space sin space x minus 8 end fraction!

D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Jawaban

nilai dari limit as x rightwards arrow infinity of fraction numerator cos space 4 x minus space 6 x over denominator 3 x plus 2 space sin space x minus 8 end fraction adalah negative 2.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah bold minus bold 2.

Ingat!

  • nilai cos space x, selalu berada di antara negative 1 dan 1
  • nilai sin space x, selalu berada di antara negative 1 dan 1
  • limit as x rightwards arrow infinity of k over x to the power of n equals 0 comma space text dimana end text space n text  bilangan asli end text
  • sifat limit: limit as x rightwards arrow 0 of left parenthesis f left parenthesis x right parenthesis colon f left parenthesis g right parenthesis right parenthesis equals limit as x rightwards arrow 0 of f left parenthesis x right parenthesis colon limit as x rightwards arrow 0 of f left parenthesis g right parenthesis

Apabila nilai x equals infinity langsung di subsitusi ke fungsi fraction numerator cos space 4 x minus space 6 x over denominator 3 x plus 2 space sin space x minus 8 end fraction maka akan di peroleh nilai limit infinity over infinity. Oleh karena itu harus disederhanakan terlebih dahulu. 

Kalikan pembilang dan penyebut dengan 1 over x, sehingga:

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of fraction numerator cos space 4 x minus space 6 x over denominator 3 x plus 2 space sin space x minus 8 end fraction end cell equals cell limit as x rightwards arrow infinity of fraction numerator cos space 4 x minus space 6 x over denominator 3 x plus 2 space sin space x minus 8 end fraction times fraction numerator begin display style bevelled 1 over x end style over denominator begin display style bevelled 1 over x end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator cos space 4 x over denominator x end fraction end style minus space begin display style fraction numerator 6 x over denominator x end fraction end style over denominator begin display style fraction numerator 3 x over denominator x end fraction end style plus begin display style fraction numerator 2 space sin space x over denominator x end fraction end style minus begin display style 8 over x end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator cos space 4 x over denominator x end fraction minus space 6 end style over denominator begin display style 3 plus fraction numerator 2 space sin space x over denominator x end fraction minus 8 over x end style end fraction end cell end table

Nilai cos space x, selalu berada di antara negative 1 dan 1 , makanegative 1 less or equal than cos space 4 x less or equal than 1, sehingga:

table attributes columnalign right center left columnspacing 0px end attributes row cell negative 1 end cell less or equal than cell cos space 4 x less or equal than 1 end cell row cell fraction numerator negative 1 over denominator x end fraction end cell less or equal than cell fraction numerator cos space 4 x over denominator x end fraction less or equal than 1 over x end cell row cell limit as x rightwards arrow infinity of fraction numerator negative 1 over denominator x end fraction end cell less or equal than cell limit as x rightwards arrow infinity of fraction numerator cos space 4 x over denominator x end fraction less or equal than limit as x rightwards arrow infinity of 1 over x end cell row 0 less or equal than cell limit as x rightwards arrow infinity of fraction numerator cos space 4 x over denominator x end fraction less or equal than 0 end cell end table

Oleh karena itu table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow infinity of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator cos space 4 x over denominator x end fraction end cell end table equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank 0 end table.

Begitu juga dengan nilai sin space x, selalu berada di antara negative 1 dan 1, maka negative 1 less or equal than sin space x less or equal than 1. Sehingga:

table attributes columnalign right center left columnspacing 0px end attributes row cell negative 1 end cell less or equal than cell sin space x less or equal than 1 end cell row cell fraction numerator negative 1 over denominator x end fraction end cell less or equal than cell fraction numerator sin space x over denominator x end fraction less or equal than 1 over x end cell row cell limit as x rightwards arrow infinity of fraction numerator negative 1 over denominator x end fraction end cell less or equal than cell limit as x rightwards arrow infinity of fraction numerator sin space x over denominator x end fraction less or equal than limit as x rightwards arrow infinity of 1 over x end cell row 0 less or equal than cell limit as x rightwards arrow infinity of fraction numerator sin space x over denominator x end fraction less or equal than 0 end cell row blank blank blank end table

Oleh karena itu table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow infinity of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator sin space x over denominator x end fraction end cell end table equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank 0 end table. Sehingga di peroleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator cos space 4 x over denominator x end fraction minus space 6 end style over denominator begin display style 3 plus fraction numerator 2 space sin space x over denominator x end fraction minus 8 over x end style end fraction end cell equals cell fraction numerator begin display style limit as x rightwards arrow infinity of fraction numerator cos space 4 x over denominator x end fraction minus limit as x rightwards arrow infinity of space 6 end style over denominator begin display style limit as x rightwards arrow infinity of 3 plus 2 limit as x rightwards arrow infinity of fraction numerator sin space x over denominator x end fraction minus limit as x rightwards arrow infinity of 8 over x end style end fraction end cell row blank equals cell fraction numerator 0 minus 6 over denominator 3 plus 0 plus 0 end fraction end cell row blank equals cell fraction numerator negative 6 over denominator 3 end fraction end cell row blank equals cell negative 2 end cell end table

Dengan demikian nilai dari limit as x rightwards arrow infinity of fraction numerator cos space 4 x minus space 6 x over denominator 3 x plus 2 space sin space x minus 8 end fraction adalah negative 2.

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