Pertanyaan

Tentukan nilai: X → ∞ lim ( x 2 − 3 x − 4 − x − 1 )

Tentukan nilai:

$X \to \infty lim (x^{2} - 3 x - 4 - x - 1)$

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

Nilai .

Nilai

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 kedua (bagi dengan suku berpangkat tertinggi dari pembilang atau penyebut): Ingat: . Jadi, Nilai .

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 .

$begin mathsize 12px style limit as straight x rightwards arrow infinity of left parenthesis square root of straight x squared minus 3 straight x minus 4 end root minus straight x minus 1 right parenthesis equals limit as straight x rightwards arrow infinity of open parentheses square root of straight x squared minus 3 straight x minus 4 end root minus left parenthesis straight x plus 1 right parenthesis close parentheses times fraction numerator open parentheses square root of straight x squared minus 3 straight x minus 4 end root plus left parenthesis straight x plus 1 right parenthesis close parentheses over denominator open parentheses square root of straight x squared minus 3 straight x minus 4 end root plus left parenthesis straight x plus 1 right parenthesis close parentheses end fraction equals limit as straight x rightwards arrow infinity of fraction numerator open parentheses square root of straight x squared minus 3 straight x minus 4 end root close parentheses squared minus left parenthesis straight x plus 1 right parenthesis squared over denominator square root of straight x squared minus 3 straight x minus 4 end root plus straight x plus 1 end fraction equals limit as straight x rightwards arrow infinity of fraction numerator straight x squared minus 3 straight x minus 4 minus left parenthesis straight x squared plus 2 straight x plus 1 right parenthesis over denominator square root of straight x squared minus 3 straight x minus 4 right parenthesis end root plus straight x plus 1 end fraction equals limit as straight x rightwards arrow infinity of fraction numerator negative 5 straight x minus 5 over denominator square root of straight x squared minus 3 straight x minus 4 right parenthesis end root plus straight x plus 1 end fraction end style$

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

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as straight x rightwards arrow infinity of fraction numerator negative 5 straight x minus 5 over denominator square root of straight x squared minus 3 straight x minus 4 end root plus straight x plus 1 end fraction end cell equals cell limit as straight x rightwards arrow infinity of fraction numerator negative 5 straight x minus 5 over denominator square root of straight x squared minus 3 straight x minus 4 end root plus straight x plus 1 end fraction cross times fraction numerator begin display style 1 over x end style over denominator begin display style 1 over x end style end fraction end cell row blank equals cell limit as straight x rightwards arrow infinity of fraction numerator begin display style fraction numerator negative 5 straight x over denominator straight x end fraction end style minus begin display style 5 over x end style over denominator square root of begin display style straight x squared over straight x squared end style minus begin display style fraction numerator 3 straight x over denominator straight x squared end fraction end style minus begin display style 4 over x squared end style end root plus begin display style straight x over straight x end style plus begin display style 1 over x end style end fraction end cell row blank equals cell limit as straight x rightwards arrow infinity of fraction numerator begin display style negative 5 minus 5 over straight x end style over denominator begin display style square root of 1 minus 3 over straight x minus 4 over straight x squared end root plus 1 plus 1 over straight x end style end fraction end cell end table$

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

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 .

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as straight x rightwards arrow infinity of fraction numerator begin display style negative 5 minus 5 over straight x end style over denominator begin display style square root of 1 minus 3 over straight x minus 4 over straight x squared end root plus 1 plus 1 over straight x end style end fraction end cell equals cell fraction numerator begin display style negative 5 minus 5 over infinity end style over denominator begin display style square root of 1 minus 3 over infinity minus 4 over infinity squared end root plus 1 plus 1 over infinity end style end fraction end cell row blank equals cell fraction numerator begin display style negative 5 minus 0 end style over denominator begin display style square root of 1 minus 0 minus 0 end root plus 1 plus 0 end style end fraction end cell row blank equals cell negative 5 over 2 end cell end table$

Jadi, Nilai .