x→∞lim​ 2x+3​−x−2​2x−1​+x+1​​=…

Pertanyaan

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 equals horizontal ellipsis  

  1. 3 plus 2 square root of 2 

  2. 3 minus 2 square root of 2 

  3. 1 plus 1 third square root of 2 

  4. 1 minus 1 third square root of 2 

  5. infinity 

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 : 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.

149

5.0 (3 rating)

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