Pertanyaan

x → ∞ lim 4 x 2 + x + 4 + x 2 + 1 3 x − 1 = …

$x \to \infty lim \frac{3 x - 1}{4 x ^{2} + x + 4 + x ^{2} + 1} = \dots$

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah E.

Pembahasan

Ingat salah satu rumus dasar limit berikut : . Diberikan bentuk limit : . Dengan mengalikan pembilang dan penyebut dengan , maka bentuk di atas dapat diubah menjadi Dengan demikian, hasil dari limit tersebut adalah . Jadi, jawaban yang tepat adalah E.

Ingat salah satu rumus dasar limit berikut : limit as x rightwards arrow infinity of space a over x to the power of n equals 0 .

Diberikan bentuk limit : $limit as x rightwards arrow infinity of space fraction numerator 3 x minus 1 over denominator square root of 4 x squared plus x plus 4 end root plus square root of x squared plus 1 end root end fraction$ .

Dengan mengalikan pembilang dan penyebut dengan 1 over x , maka bentuk di atas dapat diubah menjadi

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

Dengan demikian, hasil dari limit tersebut adalah .

Jadi, jawaban yang tepat adalah E.