Pertanyaan

Nilai dari x → ∞ lim 5 x + 7 x 2 − 2 x 3 ( 3 x 2 + x ) ( x − 3 x ) adalah ...

Nilai dari $x \to \infty lim \frac{( 3 x ^{2} + x ) ( x - 3 x )}{5 x + 7 x ^{2} - 2 x ^{3}}$ adalah ...

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Jawaban terverifikasi

Jawaban

jawaban yang benar adalah D.

Pembahasan

Untuk menyelesaikan limit menuju tak hingga , kita gunakan limit dasarnya yaitu, dengan bilangan real dan bilangan asli, artinya kita harus mengarahkan bentuk limit tak hingga menjadi rumus dasar dengan cara : Buat fungsinya menjadi bentuk pecahan, jika bentuknya dalam akar mak akalikan dengan bentuk sekawannya (merasionalkan) Bagi variabelnya dengan pangkat tertinggi Maka, Jadi,nilai dari adalah . Oleh karena itu, jawaban yang benar adalah D.

Untuk menyelesaikan limit menuju tak hingga open parentheses x rightwards arrow infinity close parentheses , kita gunakan limit dasarnya yaitu,

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

dengan bilangan real dan bilangan asli, artinya kita harus mengarahkan bentuk limit tak hingga menjadi rumus dasar dengan cara :

Buat fungsinya menjadi bentuk pecahan, jika bentuknya dalam akar mak akalikan dengan bentuk sekawannya (merasionalkan)
Bagi variabelnya dengan pangkat tertinggi

Maka,

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of fraction numerator open parentheses 3 x squared plus square root of x close parentheses open parentheses x minus 3 square root of x close parentheses over denominator 5 square root of x plus 7 x squared minus 2 x cubed end fraction end cell equals cell limit as x rightwards arrow infinity of fraction numerator 3 x cubed minus 9 x squared square root of x plus x square root of x minus 3 square root of x squared end root over denominator 5 square root of x plus 7 x squared minus 2 x cubed end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator 3 x cubed over denominator x cubed end fraction end style minus begin display style fraction numerator 9 x squared square root of x over denominator x cubed end fraction end style plus begin display style fraction numerator x square root of x over denominator x cubed end fraction end style minus begin display style fraction numerator 3 square root of x squared end root over denominator x cubed end fraction end style over denominator begin display style fraction numerator 5 square root of x over denominator x cubed end fraction end style plus begin display style fraction numerator 7 x squared over denominator x cubed end fraction end style minus begin display style fraction numerator 2 x cubed over denominator x cubed end fraction end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator 3 x cubed over denominator x cubed end fraction minus fraction numerator 9 x squared square root of x over denominator x cubed end fraction plus fraction numerator x square root of x over denominator x cubed end fraction minus fraction numerator 3 square root of x squared end root over denominator x cubed end fraction end style over denominator begin display style fraction numerator 5 square root of x over denominator x cubed end fraction plus fraction numerator 7 x squared over denominator x cubed end fraction minus fraction numerator 2 x cubed over denominator x cubed end fraction end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style 3 minus fraction numerator 9 x squared times x to the power of begin display style 1 half end style end exponent over denominator x cubed end fraction plus fraction numerator x times x to the power of 1 half end exponent over denominator x cubed end fraction minus fraction numerator 3 x over denominator x cubed end fraction end style over denominator begin display style fraction numerator 5 times x to the power of 1 half end exponent over denominator x cubed end fraction plus 7 over x minus 2 end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style 3 minus fraction numerator 9 x to the power of 2 plus begin display style 1 half end style end exponent over denominator x cubed end fraction plus x to the power of 1 plus begin display style 1 half end style end exponent over x cubed minus 3 over x squared end style over denominator begin display style fraction numerator 5 x to the power of 1 half end exponent over denominator x cubed end fraction plus 7 over x minus 2 end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style 3 minus fraction numerator 9 x to the power of begin display style 5 over 2 end style end exponent over denominator x cubed end fraction plus x to the power of begin display style 3 over 2 end style end exponent over x cubed minus 3 over x squared end style over denominator begin display style fraction numerator 5 x to the power of 1 half end exponent over denominator x cubed end fraction plus 7 over x minus 2 end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style 3 minus 9 over x to the power of begin display style 1 half end style end exponent plus x over x to the power of begin display style 3 over 2 end style end exponent minus 3 over x squared end style over denominator begin display style 5 over x to the power of begin display style 5 over 2 end style end exponent plus 7 over x minus 2 end style end fraction end cell row blank equals cell fraction numerator begin display style 3 minus 9 over infinity to the power of begin display style 1 half end style end exponent plus x over infinity to the power of begin display style 3 over 2 end style end exponent minus 3 over infinity squared end style over denominator begin display style 5 over infinity to the power of begin display style 1 half end style end exponent plus 7 over infinity minus 2 end style end fraction end cell row blank equals cell fraction numerator begin display style 3 minus 0 plus 0 minus 0 end style over denominator begin display style 0 plus 0 minus 2 end style end fraction end cell row blank equals cell fraction numerator 3 over denominator negative 2 end fraction end cell row blank equals cell negative 3 over 2 end cell end table end style$

Jadi, nilai dari $limit as x rightwards arrow infinity of fraction numerator open parentheses 3 x squared plus square root of x close parentheses open parentheses x minus 3 square root of x close parentheses over denominator 5 square root of x plus 7 x squared minus 2 x cubed end fraction$ adalah negative 3 over 2 .

Oleh karena itu, jawaban yang benar adalah D.