Pertanyaan

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

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

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Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah D.

Pembahasan

Untuk menyelesaikan bentuk limit di atas, kalikan pembilang dan penyebut masing-masing dengan . Bentuk limit tak hingga di atas dapat diselesaikan seperti berikut : Dengan demikian, hasil dari limit tak hingga di atas adalah . Jadi, jawaban yang tepat adalah D.

Untuk menyelesaikan bentuk limit di atas, kalikan pembilang dan penyebut masing-masing dengan 1 over x . Bentuk limit tak hingga di atas dapat diselesaikan seperti berikut :

$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 x squared plus x plus 1 end root minus square root of 2 x squared plus 1 end root over denominator x plus 1 plus square root of 2 x squared plus 4 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 square root of begin display style 1 over x squared end style open parentheses x squared plus x plus 1 close parentheses end root minus square root of begin display style 1 over x squared end style open parentheses 2 x squared plus 1 close parentheses end root over denominator begin display style 1 over x end style open parentheses x plus 1 close parentheses plus square root of begin display style 1 over x squared end style open parentheses 2 x squared plus 4 close parentheses 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 x squared over x squared end style plus begin display style x over x squared end style plus begin display style 1 over x squared end style end root minus square root of begin display style fraction numerator 2 x squared over denominator x squared end fraction end style plus begin display style 1 over x squared end style end root over denominator begin display style x over x end style plus begin display style 1 over x end style plus square root of begin display style fraction numerator 2 x squared over denominator x squared end fraction end style plus begin display style 4 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 square root of 1 plus begin display style 1 over x end style plus begin display style 1 over x squared end style end root minus square root of 2 plus begin display style 1 over x squared end style end root over denominator 1 plus begin display style 1 over x end style plus square root of 2 plus begin display style 4 over x squared end style end root end fraction end cell row blank equals cell fraction numerator square root of 1 plus 0 plus 0 end root minus square root of 2 plus 0 end root over denominator 1 plus 0 plus square root of 2 plus 0 end root end fraction end cell row blank equals cell fraction numerator 1 minus square root of 2 over denominator 1 plus square root of 2 end fraction times fraction numerator 1 minus square root of 2 over denominator 1 minus square root of 2 end fraction end cell row blank equals cell fraction numerator open parentheses 1 minus square root of 2 close parentheses squared over denominator 1 squared minus open parentheses square root of 2 close parentheses squared end fraction end cell row blank equals cell fraction numerator 1 minus 2 square root of 2 plus 2 over denominator 1 minus 2 end fraction end cell row blank equals cell fraction numerator 3 minus 2 square root of 2 over denominator negative 1 end fraction end cell row blank equals cell 2 square root of 2 minus 3 end cell end table$

Dengan demikian, hasil dari limit tak hingga di atas adalah 2 square root of 2 minus 3 .

Jadi, jawaban yang tepat adalah D.