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Pertanyaan

limit as x rightwards arrow infinity of fraction numerator x minus 3 over denominator square root of 1 minus 2 x plus 4 x squared end root end fraction

Pembahasan Soal:

Diketahui limit as x rightwards arrow infinity of fraction numerator x minus 3 over denominator square root of 1 minus 2 x plus 4 x squared end root end fraction. Untuk menentukan nilai limit tersebut, langkah pertama adalah membagi setiap suku dengan variabel pangkat tertinggi. Variabel pangkat tertinggi pada bentuk limit tersebut adalah x, maka membagi setiap suku dengan x sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of fraction numerator x minus 3 over denominator square root of 1 minus 2 x plus 4 x squared end root end fraction end cell equals cell limit as x rightwards arrow infinity of fraction numerator begin display style x over x end style minus begin display style 3 over x end style over denominator square root of begin display style 1 over x squared end style minus begin display style fraction numerator 2 x over denominator x squared end fraction end style plus begin display style fraction numerator 4 x squared over denominator x squared end fraction end style end root end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator 1 minus begin display style 3 over x end style over denominator square root of begin display style 1 over x squared end style minus begin display style 2 over x end style plus 4 end root end fraction end cell end table

Selanjutnya, 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, 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 x minus 3 over denominator square root of 1 minus 2 x plus 4 x squared end root end fraction end cell equals cell limit as x rightwards arrow infinity of fraction numerator 1 minus begin display style 3 over x end style over denominator square root of begin display style 1 over x squared end style minus begin display style 2 over x end style plus 4 end root end fraction end cell row blank equals cell fraction numerator 1 minus 0 over denominator square root of 0 minus 0 plus 4 end root end fraction end cell row blank equals cell fraction numerator 1 over denominator square root of 4 end fraction end cell row blank equals cell 1 half end cell end table


Dengan demikian, limit as x rightwards arrow infinity of fraction numerator x minus 3 over denominator square root of 1 minus 2 x plus 4 x squared end root end fraction equals 1 half.

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 19 Mei 2021

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