Pertanyaan

Tentukan nilai limit fungsi berikut ini! x → ∞ lim ( x − 1 ) ( x + 1 ) x 2 + x + 3

Tentukan nilai limit fungsi berikut ini!

$x \to \infty lim \frac{x ^{2} + x + 3}{( x - 1 ) ( x + 1 )}$

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

nilai adalah 1.

nilai $begin mathsize 14px style limit as x rightwards arrow infinity of space square root of fraction numerator x squared plus x plus 3 over denominator open parentheses x minus 1 close parentheses open parentheses x plus 1 close parentheses end fraction end root end style$ adalah 1.

Pembahasan

Dengan mengalikan pembilang dan penyebut dengan , diperoleh hasil perhitungan sebagai berikut. Jadi, nilai adalah 1.

Dengan mengalikan pembilang dan penyebut dengan $fraction numerator 1 over denominator square root of x squared end root end fraction$ , diperoleh hasil perhitungan sebagai berikut.

$limit as x rightwards arrow infinity of space square root of fraction numerator x squared plus x plus 3 over denominator open parentheses x minus 1 close parentheses open parentheses x plus 1 close parentheses end fraction end root equals limit as x rightwards arrow infinity of space square root of fraction numerator x squared plus x plus 3 over denominator x squared minus 1 end fraction end root equals limit as x rightwards arrow infinity of fraction numerator square root of x squared plus x plus 3 end root over denominator square root of x squared minus 1 end root end fraction equals limit as x rightwards arrow infinity of fraction numerator square root of x squared plus x plus 3 end root over denominator square root of x squared minus 1 end root end fraction cross times fraction numerator begin display style fraction numerator 1 over denominator square root of x squared end root end fraction end style over denominator fraction numerator 1 over denominator square root of x squared end root end fraction end fraction equals limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator square root of x squared plus x plus 3 end root over denominator square root of x squared end root end fraction end style over denominator begin display style fraction numerator square root of x squared minus 1 end root over denominator square root of x squared end root end fraction end style end fraction equals limit as x rightwards arrow infinity of 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 3 over x squared end style end root over denominator square root of begin display style x squared over x squared end style minus begin display style 1 over x squared end style end root end fraction equals limit as x rightwards arrow infinity of fraction numerator square root of 1 plus begin display style 1 over x end style plus begin display style 3 over x squared end style end root over denominator square root of 1 minus 1 over x squared end root end fraction equals fraction numerator square root of 1 plus 0 plus 0 end root over denominator square root of 1 minus 0 end root end fraction equals fraction numerator square root of 1 over denominator square root of 1 end fraction equals 1$

Jadi, nilai $begin mathsize 14px style limit as x rightwards arrow infinity of space square root of fraction numerator x squared plus x plus 3 over denominator open parentheses x minus 1 close parentheses open parentheses x plus 1 close parentheses end fraction end root end style$ adalah 1.

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