Pertanyaan

Nilai dari: x → ∞ lim 4 x 2 − 12 x − 1 − 2 x + 1 = ....

Nilai dari: $x \to \infty lim 4 x^{2} - 12 x - 1 - 2 x + 1 = ....$

N. Puspita

Master Teacher

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Nilai .

Nilai

Pembahasan

Konsep limit tak hingga: metode membagi dengan pangkat tertinggi Diketahui: Fungsi di atas belum fungsi bentuk rasional,, dengan perkalian akar sekawan maka: Pangkat tertinggi adalah atau sama dengan , maka pembilang dan penyebut dibagi oleh , maka diperoleh penyelesaiannyasebagai berikut: Jadi, Nilai .

Konsep limit tak hingga: metode membagi dengan pangkat tertinggi

Diketahui:

Fungsi di atas belum fungsi bentuk rasional,, dengan perkalian akar sekawan maka:

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as straight x rightwards arrow infinity of square root of 4 straight x squared minus 12 straight x minus 1 end root minus 2 straight x plus 1 end cell row blank equals cell space limit as straight x rightwards arrow infinity of square root of 4 straight x squared minus 12 straight x minus 1 end root minus left parenthesis 2 straight x minus 1 right parenthesis end cell row blank equals cell limit as straight x rightwards arrow infinity of square root of 4 straight x squared minus 12 straight x minus 1 end root minus left parenthesis 2 straight x minus 1 right parenthesis times fraction numerator square root of 4 straight x squared minus 12 straight x minus 1 end root plus left parenthesis 2 straight x minus 1 right parenthesis over denominator square root of 4 straight x squared minus 12 straight x minus 1 end root plus left parenthesis 2 straight x minus 1 right parenthesis end fraction end cell row blank equals cell limit as straight x rightwards arrow infinity of fraction numerator 4 straight x squared minus 12 straight x minus 1 minus left parenthesis 2 straight x minus 1 right parenthesis squared over denominator square root of 4 straight x squared minus 12 straight x minus 1 end root plus left parenthesis 2 straight x minus 1 right parenthesis end fraction end cell row blank equals cell limit as straight x rightwards arrow infinity of fraction numerator 4 straight x squared minus 12 straight x minus 1 minus left parenthesis 4 straight x squared minus 4 straight x plus 1 right parenthesis over denominator square root of 4 straight x squared minus 12 straight x minus 1 end root plus left parenthesis 2 straight x minus 1 right parenthesis end fraction end cell row blank equals cell limit as straight x rightwards arrow infinity of fraction numerator 4 straight x squared minus 12 straight x minus 1 minus 4 straight x squared plus 4 straight x minus 1 over denominator square root of 4 straight x squared minus 12 straight x minus 1 end root plus left parenthesis 2 straight x minus 1 right parenthesis end fraction end cell row blank equals cell limit as straight x rightwards arrow infinity of fraction numerator negative 8 straight x minus 2 over denominator square root of 4 straight x squared minus 12 straight x minus 1 end root plus left parenthesis 2 straight x minus 1 right parenthesis end fraction end cell end table end style$

Pangkat tertinggi adalah atau sama dengan , maka pembilang dan penyebut dibagi oleh , maka diperoleh penyelesaiannya sebagai berikut:

$begin mathsize 11px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as straight x rightwards arrow infinity of fraction numerator negative 8 straight x minus 2 over denominator square root of 4 straight x squared minus 12 straight x minus 1 end root plus left parenthesis 2 straight x minus 1 right parenthesis end fraction end cell equals cell limit as straight x rightwards arrow infinity of fraction numerator begin display style fraction numerator negative 8 straight x over denominator straight x end fraction minus 2 over straight x end style over denominator square root of begin display style fraction numerator 4 straight x squared over denominator straight x squared end fraction minus fraction numerator 12 straight x over denominator straight x squared end fraction minus 1 over straight x squared end style end root plus left parenthesis begin display style fraction numerator 2 straight x over denominator straight x end fraction end style minus begin display style 1 over straight x end style right parenthesis end fraction end cell row blank equals cell limit as straight x rightwards arrow infinity of fraction numerator begin display style negative 8 minus 2 over straight x end style over denominator square root of begin display style 4 minus 12 over straight x minus 1 over straight x squared end style end root plus left parenthesis begin display style 2 end style minus begin display style 1 over straight x end style right parenthesis end fraction end cell row blank equals cell fraction numerator begin display style negative 8 minus 2 over infinity end style over denominator square root of begin display style 4 minus 12 over infinity minus 1 over infinity squared end style end root plus left parenthesis begin display style 2 end style minus begin display style 1 over infinity end style right parenthesis end fraction end cell row blank equals cell fraction numerator begin display style negative 8 minus 0 end style over denominator square root of begin display style 4 minus 0 minus 0 end style end root plus left parenthesis begin display style 2 end style minus begin display style 0 end style right parenthesis end fraction end cell row blank equals cell fraction numerator negative 8 over denominator 2 plus 2 end fraction end cell row blank equals cell negative 2 end cell end table end style$

Jadi, Nilai .

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