Pertanyaan

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

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

N. Puspita

Master Teacher

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Pembahasan

Langkah-langkah limit tak hingga bentuk akar: Kali dengan akar sekawan. Bagi dengan suku berpangkat tertinggi dari pembilang atau penyebut. Substitusi x mendekati Langkah pertama (kali dengan akar sekawan): Ingat: . Langkah kedua (bagi dengan suku berpangkat tertinggi dari pembilang atau penyebut): Langkah ketiga (substitusi x mendekati ): Ingat: . Jadi, nilai dari .

Langkah-langkah limit tak hingga bentuk akar:

Kali dengan akar sekawan.
Bagi dengan suku berpangkat tertinggi dari pembilang atau penyebut.
Substitusi x mendekati

Langkah pertama (kali dengan akar sekawan):

Ingat: left parenthesis a minus b right parenthesis left parenthesis a plus b right parenthesis equals a squared minus b squared .

$limit as x rightwards arrow infinity of square root of straight x squared plus 4 straight x minus 1 end root minus square root of straight x squared minus 4 straight x plus 2 end root equals limit as x rightwards arrow infinity of open parentheses square root of straight x squared plus 4 straight x minus 1 end root minus square root of straight x squared minus 4 straight x plus 2 end root close parentheses cross times fraction numerator open parentheses square root of straight x squared plus 4 straight x minus 1 end root plus square root of straight x squared minus 4 straight x plus 2 end root close parentheses over denominator open parentheses square root of straight x squared plus 4 straight x minus 1 end root plus square root of straight x squared minus 4 straight x plus 2 end root close parentheses end fraction equals limit as x rightwards arrow infinity of fraction numerator open parentheses square root of straight x squared plus 4 straight x minus 1 end root close parentheses squared minus open parentheses square root of straight x squared minus 4 straight x plus 2 end root close parentheses squared over denominator square root of straight x squared plus 4 straight x minus 1 end root plus square root of straight x squared minus 4 straight x plus 2 end root end fraction equals limit as x rightwards arrow infinity of fraction numerator straight x squared plus 4 straight x minus 1 minus straight x squared plus 4 straight x minus 2 over denominator square root of straight x squared plus 4 straight x minus 1 end root plus square root of straight x squared minus 4 straight x plus 2 end root end fraction equals limit as x rightwards arrow infinity of fraction numerator 8 straight x minus 3 over denominator square root of straight x squared plus 4 straight x minus 1 end root plus square root of straight x squared minus 4 straight x plus 2 end root end fraction$

Langkah kedua (bagi dengan suku berpangkat tertinggi dari pembilang atau penyebut):

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of fraction numerator 8 straight x minus 3 over denominator square root of straight x squared plus 4 straight x minus 1 end root plus square root of straight x squared minus 4 straight x plus 2 end root end fraction end cell equals cell limit as x rightwards arrow infinity of fraction numerator 8 straight x minus 3 over denominator square root of straight x squared plus 4 straight x minus 1 end root plus square root of straight x squared minus 4 straight x plus 2 end root end fraction cross times fraction numerator begin display style 1 over x end style over denominator begin display style 1 over x end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator 8 straight x over denominator straight x end fraction end style minus begin display style 3 over x end style over denominator square root of begin display style straight x squared over straight x squared end style plus begin display style fraction numerator 4 straight x over denominator straight x squared end fraction end style minus begin display style 1 over straight x squared end style end root plus square root of begin display style straight x squared over straight x squared end style minus begin display style fraction numerator 4 straight x over denominator straight x squared end fraction end style plus begin display style 2 over straight x squared end style end root end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style 8 minus 3 over x end style over denominator square root of begin display style 1 plus 4 over straight x minus 1 over straight x squared end style end root plus square root of begin display style 1 minus 4 over straight x plus 2 over straight x squared end style end root end fraction end cell end table$

Langkah ketiga (substitusi x mendekati infinity ):

Ingat: limit as x rightwards arrow infinity of k over x to the power of n equals k over infinity to the power of n equals 0 .

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of fraction numerator begin display style 8 minus 3 over x end style over denominator square root of begin display style 1 plus 4 over straight x minus 1 over straight x squared end style end root plus square root of begin display style 1 minus 4 over straight x plus 2 over straight x squared end style end root end fraction end cell equals cell fraction numerator begin display style 8 minus 3 over infinity end style over denominator square root of begin display style 1 plus 4 over infinity minus 1 over infinity squared end style end root plus square root of begin display style 1 minus 4 over infinity plus 2 over infinity squared end style end root end fraction end cell row blank equals cell fraction numerator begin display style 8 minus 0 end style over denominator square root of begin display style 1 plus 0 minus 0 end style end root plus square root of begin display style 1 minus 0 plus 0 end style end root end fraction end cell row blank equals cell 8 over 2 end cell row blank equals 4 end table$

Jadi, nilai dari .