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Diketahui fungsi f(x)=3x5+2x2−3x+1 dan g(x)=x3+3x2−2x−5. Hasil x→∞lim​g(x)f(x)​ adalah ....

Pertanyaan

Diketahui fungsi f left parenthesis x right parenthesis equals 3 x to the power of 5 plus 2 x squared minus 3 x plus 1 dan g left parenthesis x right parenthesis equals x cubed plus 3 x squared minus 2 x minus 5. Hasil limit as x rightwards arrow infinity of fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction adalah ....

  1. negative infinity

  2. negative 3

  3. 0

  4. 3

     

  5. infinity

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah E.

Pembahasan

Jawaban yang benar untuk pertanyaan tesebut adalah E.

Soal di atas merupakan soal limit tak hingga dari hasil bagi dua fungsi. Jika diselesaikan dengan menggunakan metode substitusi, maka diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction end cell equals cell limit as x rightwards arrow infinity of fraction numerator 3 x to the power of 5 plus 2 x squared minus 3 x plus 1 over denominator x cubed plus 3 x squared minus 2 x minus 5 end fraction end cell row blank equals cell fraction numerator infinity plus infinity minus infinity plus 1 over denominator infinity plus infinity minus infinity minus 5 end fraction end cell row blank equals cell fraction numerator infinity minus infinity over denominator infinity minus infinity end fraction end cell end table

Karena table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell infinity minus infinity end cell end table tidak tentu, maka soal tersebut perlu diselesaikan dengan cara lain.

Ingat bahwa pada limit tak hingga, berlaku:

limit as x rightwards arrow infinity of c over x to the power of n equals 0

Diperoleh,

limit as x rightwards arrow infinity 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 infinity of fraction numerator 3 x to the power of 5 plus 2 x squared minus 3 x plus 1 over denominator x cubed plus 3 x squared minus 2 x minus 5 end fraction equals limit as x rightwards arrow infinity of fraction numerator 3 x to the power of 5 plus 2 x squared minus 3 x plus 1 over denominator x cubed plus 3 x squared minus 2 x minus 5 end fraction times fraction numerator begin display style x to the power of 5 over x to the power of 5 end style over denominator begin display style x cubed over x cubed end style end fraction equals limit as x rightwards arrow infinity of fraction numerator x to the power of 5 open parentheses begin display style fraction numerator 3 x to the power of 5 plus 2 x squared minus 3 x plus 1 over denominator x to the power of 5 end fraction end style close parentheses over denominator x cubed open parentheses begin display style fraction numerator x cubed plus 3 x squared minus 2 x minus 5 over denominator x cubed end fraction end style close parentheses end fraction equals limit as x rightwards arrow infinity of fraction numerator x to the power of 5 open parentheses 3 plus begin display style 2 over x cubed end style minus begin display style 3 over x to the power of 4 end style plus begin display style 1 over x to the power of 5 end style close parentheses over denominator x cubed open parentheses 1 plus begin display style 3 over x end style minus begin display style 2 over x squared end style minus begin display style 5 over x cubed end style close parentheses end fraction equals limit as x rightwards arrow infinity of fraction numerator x cubed times x squared over denominator x cubed end fraction times fraction numerator limit as x rightwards arrow infinity of 3 plus limit as x rightwards arrow infinity of begin display style 2 over x cubed end style minus limit as x rightwards arrow infinity of begin display style 3 over x to the power of 4 end style plus limit as x rightwards arrow infinity of begin display style 1 over x to the power of 5 end style over denominator limit as x rightwards arrow infinity of 1 plus limit as x rightwards arrow infinity of begin display style 3 over x end style minus limit as x rightwards arrow infinity of begin display style 2 over x squared end style minus limit as x rightwards arrow infinity of begin display style 5 over x cubed end style end fraction equals limit as x rightwards arrow infinity of x squared times fraction numerator 3 plus 0 minus 0 plus 0 over denominator 1 plus 0 minus 0 minus 0 end fraction equals infinity times 3 over 1 equals infinity

Dengan demikain, hasil limit as x rightwards arrow infinity of fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction adalah infinity.

Oleh karena itu, jawaban yang benar adalah E.

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