Pertanyaan

Nilai dari x → ∞ lim 4 x 3 + 2 x 2 − x + 3 x 2 + 3 x + 1 adalah ...

Nilai dari $x \to \infty lim \frac{x ^{2} + 3 x + 1}{4 x ^{3} + 2 x ^{2} - x + 3}$ adalah ...

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Jawaban terverifikasi

Jawaban

dapat disimpulkan bahwa nilaidari adalah .

dapat disimpulkan bahwa nilai dari $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow infinity thin space of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses fraction numerator x squared plus 3 x plus 1 over denominator 4 x cubed plus 2 x squared minus x plus 3 end fraction close parentheses end cell end table$ adalah

Pembahasan

Langkah pertama untuk mengetahui nilai dari maka lakukan substitusi biasa terlebih dahulu. Karena hasil substitusi biasa yaitu bentuk tak tentu, maka untuk mengetahui limit tak hingga tersebut, dapat dengan memanipulasi aljabar sebagai berikut: Jadi, dapat disimpulkan bahwa nilaidari adalah .

Langkah pertama untuk mengetahui nilai dari $limit as x rightwards arrow infinity of fraction numerator x squared plus 3 x plus 1 over denominator 4 x cubed plus 2 x squared minus x plus 3 end fraction$ maka lakukan substitusi biasa terlebih dahulu.

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of fraction numerator x squared plus 3 x plus 1 over denominator 4 x cubed plus 2 x squared minus x plus 3 end fraction end cell equals cell fraction numerator infinity squared plus 3 times infinity plus 1 over denominator 4 times infinity cubed plus 2 times infinity squared minus infinity plus 3 end fraction end cell row blank equals cell fraction numerator infinity plus infinity plus 1 over denominator infinity plus infinity minus infinity plus 3 end fraction end cell row blank equals cell infinity over infinity end cell row blank equals cell tak space tentu end cell end table$

Karena hasil substitusi biasa yaitu bentuk tak tentu, maka untuk mengetahui limit tak hingga tersebut, dapat dengan memanipulasi aljabar sebagai berikut:

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity thin space of open parentheses fraction numerator x squared plus 3 x plus 1 over denominator 4 x cubed plus 2 x squared minus x plus 3 end fraction close parentheses end cell equals cell limit as x rightwards arrow infinity thin space of open parentheses fraction numerator x squared plus 3 x plus 1 over denominator 4 x cubed plus 2 x squared minus x plus 3 end fraction close parentheses colon x cubed over x cubed end cell row blank equals cell limit as x rightwards arrow infinity thin space of open parentheses fraction numerator begin display style x squared over x cubed plus fraction numerator 3 x over denominator x cubed end fraction plus 1 over x cubed end style over denominator begin display style fraction numerator 4 x cubed over denominator x cubed end fraction end style plus begin display style fraction numerator 2 x squared over denominator x cubed end fraction end style minus begin display style x over x cubed end style plus begin display style 3 over x cubed end style end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow infinity thin space of open parentheses fraction numerator begin display style 1 over x end style plus begin display style 3 over x squared end style plus begin display style 1 over x cubed end style over denominator 4 plus begin display style 2 over x end style minus begin display style 1 over x squared end style plus begin display style 3 over x cubed end style end fraction close parentheses end cell row blank equals cell fraction numerator begin display style 1 over infinity plus 3 over infinity squared plus 1 over infinity cubed end style over denominator 4 plus begin display style 2 over infinity end style minus begin display style 1 over infinity squared end style plus begin display style 3 over infinity cubed end style end fraction end cell row blank equals cell fraction numerator 0 plus 0 plus 0 over denominator 4 plus 0 minus 0 plus 0 end fraction end cell row blank equals cell 0 over 4 end cell row blank equals 0 end table$

Jadi, dapat disimpulkan bahwa nilai dari $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow infinity thin space of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses fraction numerator x squared plus 3 x plus 1 over denominator 4 x cubed plus 2 x squared minus x plus 3 end fraction close parentheses end cell end table$ adalah .