Pertanyaan

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

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

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Habis dalam

Klaim

I. Kumaralalita

Master Teacher

Mahasiswa/Alumni Universitas Gadjah Mada

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah A.

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 A.

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 : $limit as x rightwards arrow infinity of space fraction numerator square root of 2 x minus 1 end root plus square root of x plus 1 end root over denominator square root of 2 x plus 3 end root minus square root of x minus 2 end root end fraction$ .

Dengan mengalikan pembilang dan penyebut dengan square root of 1 over x end root , 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 square root of 2 x minus 1 end root plus square root of x plus 1 end root over denominator square root of 2 x plus 3 end root minus square root of x minus 2 end root end fraction times fraction numerator square root of begin display style 1 over x end style end root over denominator square root of begin display style 1 over x end style end root end fraction end cell row blank equals cell limit as x rightwards arrow infinity of space fraction numerator square root of begin display style 1 over x end style open parentheses 2 x minus 1 close parentheses end root plus square root of begin display style 1 over x end style open parentheses x plus 1 close parentheses end root over denominator square root of begin display style 1 over x end style left parenthesis 2 x plus 3 right parenthesis end root minus square root of begin display style 1 over x end style left parenthesis x minus 2 right parenthesis end root end fraction end cell row blank equals cell limit as x rightwards arrow infinity of space fraction numerator square root of begin display style fraction numerator 2 x over denominator x end fraction end style minus begin display style 1 over x end style end root plus square root of begin display style x over x end style plus begin display style 1 over x end style end root over denominator square root of begin display style fraction numerator 2 x over denominator x end fraction end style plus begin display style 3 over x end style end root minus square root of begin display style x over x end style minus begin display style 2 over x end style end root end fraction end cell row blank equals cell limit as x rightwards arrow infinity of space fraction numerator square root of 2 minus begin display style 1 over x end style end root plus square root of 1 plus begin display style 1 over x end style end root over denominator square root of 2 plus begin display style 3 over x end style end root minus square root of 1 minus begin display style 2 over x end style end root end fraction end cell row blank equals cell fraction numerator square root of 2 minus 0 end root plus square root of 1 plus 0 end root over denominator square root of 2 plus 0 end root minus square root of 1 minus 0 end root end fraction end cell row blank equals cell fraction numerator square root of 2 plus 1 over denominator square root of 2 minus 1 end fraction times fraction numerator square root of 2 plus 1 over denominator square root of 2 plus 1 end fraction end cell row blank equals cell fraction numerator open parentheses square root of 2 plus 1 close parentheses squared over denominator open parentheses square root of 2 close parentheses squared minus 1 squared end fraction end cell row blank equals cell fraction numerator 2 plus 2 square root of 2 plus 1 over denominator 2 minus 1 end fraction end cell row blank equals cell 3 plus 2 square root of 2 end cell end table$

Dengan demikian, hasil dari limit tersebut adalah 3 plus 2 square root of 2 .

Jadi, jawaban yang tepat adalah A.