Pertanyaan

Nilai x → ∞ lim ( 4 x 2 + 3 x + 3 − 4 x 2 − 5 x + 3 ) = ... .

Nilai $x \to \infty lim (4 x^{2} + 3 x + 3 - 4 x^{2} - 5 x + 3) = ... .$

N. Puspita

Master Teacher

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nilai .

nilai

limit as x rightwards arrow infinity of open parentheses square root of 4 x squared plus 3 x plus 3 end root minus square root of 4 x squared minus 5 x plus 3 end root close parentheses equals 2

Pembahasan

Diketahui . Langkah pertama untuk menentukan nilai limit tersebut adalah mengalikan dengan sekawan sebagai berikut. Kemudian, membagi setiap suku dengan variabel pangkat tertinggi. Variabel pangkat tertinggi adalah maka membagi setiap suku dengan sebagai berikut. Selanjutnya, dengan menggunakan teorema, jika bilangan asli maka , nilai limit tersebut sebagai berikut. Dengan demikian, nilai .

Diketahui limit as x rightwards arrow infinity of open parentheses square root of 4 x squared plus 3 x plus 3 end root minus square root of 4 x squared minus 5 x plus 3 end root close parentheses . Langkah pertama untuk menentukan nilai limit tersebut adalah mengalikan dengan sekawan sebagai berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow infinity of open parentheses square root of 4 x squared plus 3 x plus 3 end root minus square root of 4 x squared minus 5 x plus 3 end root close parentheses end cell row blank equals cell limit as x rightwards arrow infinity of open parentheses square root of 4 x squared plus 3 x plus 3 end root minus square root of 4 x squared minus 5 x plus 3 end root close parentheses times end cell row blank blank cell fraction numerator open parentheses square root of 4 x squared plus 3 x plus 3 end root plus square root of 4 x squared minus 5 x plus 3 end root close parentheses over denominator open parentheses square root of 4 x squared plus 3 x plus 3 end root plus square root of 4 x squared minus 5 x plus 3 end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator open parentheses square root of 4 x squared plus 3 x plus 3 end root close parentheses squared minus open parentheses square root of 4 x squared minus 5 x plus 3 end root close parentheses squared over denominator open parentheses square root of 4 x squared plus 3 x plus 3 end root plus square root of 4 x squared minus 5 x plus 3 end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator 4 x squared plus 3 x plus 3 minus open parentheses 4 x squared minus 5 x plus 3 close parentheses over denominator open parentheses square root of 4 x squared plus 3 x plus 3 end root plus square root of 4 x squared minus 5 x plus 3 end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator 4 x squared minus 4 x squared plus 3 x plus 5 x plus 3 minus 3 over denominator open parentheses square root of 4 x squared plus 3 x plus 3 end root plus square root of 4 x squared minus 5 x plus 3 end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator 8 x over denominator open parentheses square root of 4 x squared plus 3 x plus 3 end root plus square root of 4 x squared minus 5 x plus 3 end root close parentheses end fraction end cell end table$

Kemudian, membagi setiap suku dengan variabel pangkat tertinggi. Variabel pangkat tertinggi adalah maka membagi setiap suku dengan sebagai berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow infinity of open parentheses square root of 4 x squared plus 3 x plus 3 end root minus square root of 4 x squared minus 5 x plus 3 end root close parentheses end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator 8 x over denominator square root of 4 x squared plus 3 x plus 3 end root plus square root of 4 x squared minus 5 x plus 3 end root end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator 8 x over denominator x end fraction end style over denominator square root of begin display style fraction numerator 4 x squared over denominator x squared end fraction end style plus begin display style fraction numerator 3 x over denominator x squared end fraction end style plus begin display style 3 over x squared end style end root plus square root of begin display style fraction numerator 4 x squared over denominator x squared end fraction end style minus begin display style fraction numerator 5 x over denominator x squared end fraction end style plus begin display style 3 over x squared end style end root end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator 8 over denominator square root of 4 plus begin display style 3 over x end style plus begin display style 3 over x squared end style end root plus square root of 4 minus begin display style 5 over x end style plus begin display style 3 over x squared end style end root end fraction end cell end table$

Selanjutnya, dengan menggunakan teorema, jika bilangan asli maka limit as x rightwards arrow infinity of 1 over x to the power of k equals 0 , nilai limit tersebut sebagai berikut.

Dengan demikian, nilai limit as x rightwards arrow infinity of open parentheses square root of 4 x squared plus 3 x plus 3 end root minus square root of 4 x squared minus 5 x plus 3 end root close parentheses equals 2 .

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