Nilai x→∞lim5x−4x23x2−2x−6=....

Pertanyaan

Nilai $\lim_{x\rightarrow\infty}\frac{3x^2-2x-6}{5x-4x^2}=...\;.$

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Pembahasan Soal:

Langkah pertama untuk menentukan nilai limit $limit as x rightwards arrow infinity of fraction numerator 3 x squared minus 2 x minus 6 over denominator 5 x minus 4 x squared end fraction$ adalah membagi setiap suku dengan variabel pangkat tertinggi. Variabel pangkat tertinggi pada bentuk limit tersebut adalah $x squared$ , maka diperoleh:

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of fraction numerator 3 x squared minus 2 x minus 6 over denominator 5 x minus 4 x squared end fraction end cell equals cell limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator 3 x squared over denominator x squared end fraction end style minus begin display style fraction numerator 2 x over denominator x squared end fraction end style minus begin display style 6 over x squared end style over denominator begin display style fraction numerator 5 x over denominator x squared end fraction end style minus begin display style fraction numerator 4 x squared over denominator x squared end fraction end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator 3 minus begin display style 2 over x end style minus begin display style 6 over x squared end style over denominator begin display style 5 over x end style minus 4 end fraction end cell end table$

Kemudian, dengan menggunakan teorema jika $k$ bilangan asli maka $limit as x rightwards arrow infinity of 1 over x to the power of k equals 0$ , diperoleh nilai limit tersebut sebagai berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of fraction numerator 3 x squared minus 2 x minus 6 over denominator 5 x minus 4 x squared end fraction end cell equals cell limit as x rightwards arrow infinity of fraction numerator 3 minus begin display style 2 over x end style minus begin display style 6 over x squared end style over denominator begin display style 5 over x end style minus 4 end fraction end cell row blank equals cell fraction numerator 3 minus 0 minus 0 over denominator 0 minus 4 end fraction end cell row blank equals cell fraction numerator 3 over denominator negative 4 end fraction end cell row blank equals cell negative 3 over 4 end cell end table$

Dengan demikian, nilai $limit as x rightwards arrow infinity of fraction numerator 3 x squared minus 2 x minus 6 over denominator 5 x minus 4 x squared end fraction equals negative 3 over 4$ .

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Terakhir diupdate 07 Oktober 2021