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x → ∞ lim ​ ( 36 x 2 + 12 x − 3 ​ − 5 x − 2 − x 2 + 18 x + 3 ​ ) = ....

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N. Puspita

Master Teacher

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limit as x rightwards arrow infinity of open parentheses square root of 36 x squared plus 12 x minus 3 end root minus 5 x minus 2 minus square root of x squared plus 18 x plus 3 end root close parentheses equals negative 10.

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Rumus dasar limit tak hingga sebagai berikut. Kita pecah menjadi dua limit untuk mengerjakan agar tidak terlalu panjang. Limit pertama Limit kedua Kemudian kita jumlahkan hasilnya. Jadi, .

Rumus dasar limit tak hingga sebagai berikut.

limit as x rightwards arrow infinity of a over x to the power of n equals 0

Kita pecah menjadi dua limit untuk mengerjakan agar tidak terlalu panjang.

begin mathsize 11px style limit as x rightwards arrow infinity of open parentheses square root of 36 x squared plus 12 x minus 3 end root minus 5 x minus 2 minus square root of x squared plus 18 x plus 3 end root close parentheses equals limit as x rightwards arrow infinity of open parentheses square root of 36 x squared plus 12 x minus 3 end root minus 6 x plus x minus 2 plus negative square root of x squared plus 18 x plus 3 end root close parentheses equals limit as x rightwards arrow infinity of square root of 36 x squared plus 12 x minus 3 end root minus 6 x plus limit as x rightwards arrow infinity of x minus 2 minus square root of x squared plus 18 x plus 3 end root end style

Limit pertama

begin mathsize 11px style limit as x rightwards arrow infinity of square root of 36 x squared plus 12 x minus 3 end root minus 6 x equals limit as x rightwards arrow infinity of square root of 36 x squared plus 12 x minus 3 end root minus square root of open parentheses 6 x close parentheses squared end root equals limit as x rightwards arrow infinity of square root of 36 x squared plus 12 x minus 3 end root minus square root of 36 x squared end root equals limit as x rightwards arrow infinity of square root of 36 x squared plus 12 x minus 3 end root minus square root of 36 x squared end root times fraction numerator square root of 36 x squared plus 12 x minus 3 end root plus square root of 36 x squared end root over denominator square root of 36 x squared plus 12 x minus 3 end root plus square root of 36 x squared end root end fraction equals limit as x rightwards arrow infinity of fraction numerator 36 x squared plus 12 x minus 3 minus open parentheses 36 x squared close parentheses over denominator square root of 36 x squared plus 12 x minus 3 end root plus square root of 36 x squared end root end fraction equals limit as x rightwards arrow infinity of fraction numerator 12 x minus 3 over denominator square root of 36 x squared plus 12 x minus 3 end root plus square root of 36 x squared end root end fraction. fraction numerator begin display style 1 over x end style over denominator begin display style 1 over x end style end fraction equals limit as x rightwards arrow infinity of fraction numerator begin display style 12 minus 3 over x end style over denominator begin display style square root of 36 plus 12 over x minus 3 over x squared end root plus square root of 36 end style end fraction equals fraction numerator 12 over denominator square root of 36 plus square root of 36 end fraction equals fraction numerator 12 over denominator 6 plus 6 end fraction equals 1 end style

Limit kedua

begin mathsize 11px style limit as x rightwards arrow infinity of x minus 2 minus square root of x squared plus 18 x plus 3 end root equals limit as x rightwards arrow infinity of square root of open parentheses x minus 2 close parentheses squared end root minus square root of x squared plus 18 x plus 3 end root equals limit as x rightwards arrow infinity of square root of x squared minus 4 x plus 4 end root minus square root of x squared plus 18 x plus 3 end root times fraction numerator square root of x squared minus 4 x plus 4 end root plus square root of x squared plus 18 x plus 3 end root over denominator square root of x squared minus 4 x plus 4 end root plus square root of x squared plus 18 x plus 3 end root end fraction equals limit as x rightwards arrow infinity of fraction numerator x squared minus 4 x plus 4 minus open parentheses x squared plus 18 x plus 3 close parentheses over denominator square root of x squared minus 4 x plus 4 end root plus square root of x squared plus 18 x plus 3 end root end fraction equals limit as x rightwards arrow infinity of fraction numerator negative 22 x plus 1 over denominator square root of x squared minus 4 x plus 4 end root plus square root of x squared plus 18 x plus 3 end root end fraction. fraction numerator begin display style 1 over x end style over denominator begin display style 1 over x end style end fraction equals limit as x rightwards arrow infinity of fraction numerator begin display style negative 22 plus 1 over x end style over denominator begin display style square root of 1 minus 4 over x plus 4 over x squared end root plus square root of 1 plus 18 over x plus 3 over x squared end root end style end fraction equals fraction numerator negative 22 over denominator square root of 1 plus square root of 1 end fraction equals fraction numerator negative 22 over denominator 2 end fraction equals negative 11 end style

Kemudian kita jumlahkan hasilnya.

begin mathsize 11px style limit as x rightwards arrow infinity of square root of 36 x squared plus 12 x minus 3 end root minus 6 x plus limit as x rightwards arrow infinity of x minus 2 minus square root of x squared plus 18 x plus 3 end root equals 1 plus open parentheses negative 11 close parentheses equals negative 10 end style

Jadi, limit as x rightwards arrow infinity of open parentheses square root of 36 x squared plus 12 x minus 3 end root minus 5 x minus 2 minus square root of x squared plus 18 x plus 3 end root close parentheses equals negative 10.

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