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Tentukan nilai: X→∞lim​(x2−3x−4​−x−1)

Pertanyaan

Tentukan nilai:

begin mathsize 14px 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 end style 

Pembahasan Soal:

Langkah-langkah limit tak hingga bentuk akar:

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

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 begin mathsize 14px 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 negative 5 over 2 end style.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

A. Acfreelance

Terakhir diupdate 07 Oktober 2021

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Pertanyaan yang serupa

x→∞lim​2x−36x2+5x+1​​=...

Pembahasan Soal:

0

Roboguru

Tentukan nilai: X→∞lim​(9x2+5​−4x2−6​x2−4​+x2+3​​)

Pembahasan Soal:

Dengan menerapkan konsep limit fungsi mendekati takhingga.

Variabel begin mathsize 14px style straight x end style dengan pangkat terbesar dari begin mathsize 14px style fraction numerator square root of straight x squared minus 4 end root plus square root of straight x squared plus 3 end root over denominator square root of 9 straight x squared plus 5 end root minus square root of 4 straight x squared minus 6 end root end fraction end style adalah begin mathsize 14px style square root of straight x squared end root end style maka pembilang dan penyebut pada limit fungsi dibagi oleh begin mathsize 14px style square root of straight x squared end root end style. Sehingga diperoleh:

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as straight x rightwards arrow infinity of left parenthesis fraction numerator square root of straight x squared minus 4 end root plus square root of straight x squared plus 3 end root over denominator square root of 9 straight x squared plus 5 end root minus square root of 4 straight x squared minus 6 end root end fraction right parenthesis end cell equals cell limit as straight x rightwards arrow infinity of left parenthesis fraction numerator begin display style fraction numerator square root of straight x squared minus 4 end root over denominator square root of straight x squared end root end fraction plus fraction numerator square root of straight x squared plus 3 end root over denominator square root of straight x squared end root end fraction end style over denominator begin display style fraction numerator square root of 9 straight x squared plus 5 end root over denominator square root of straight x squared end root end fraction minus fraction numerator square root of 4 straight x squared minus 6 end root over denominator square root of straight x squared end root end fraction end style end fraction right parenthesis end cell row blank equals cell limit as straight x rightwards arrow infinity of left parenthesis fraction numerator begin display style square root of fraction numerator straight x squared minus 4 over denominator straight x squared end fraction end root plus square root of fraction numerator begin display style straight x squared plus 3 end style over denominator begin display style straight x squared end style end fraction end root end style over denominator begin display style square root of fraction numerator 9 straight x squared plus 5 over denominator straight x squared end fraction end root minus square root of fraction numerator begin display style 4 straight x squared minus 6 end style over denominator begin display style straight x squared end style end fraction end root end style end fraction right parenthesis end cell row blank equals cell limit as straight x rightwards arrow infinity of left parenthesis fraction numerator begin display style square root of 1 minus 4 over straight x squared end root plus square root of 1 plus fraction numerator begin display style 3 end style over denominator begin display style straight x squared end style end fraction end root end style over denominator begin display style square root of 9 plus 5 over straight x squared end root minus square root of 4 minus fraction numerator begin display style 6 end style over denominator begin display style straight x squared end style end fraction end root end style end fraction right parenthesis end cell row blank equals cell left parenthesis fraction numerator begin display style square root of 1 minus 4 over infinity squared end root plus square root of 1 plus fraction numerator begin display style 3 end style over denominator begin display style infinity squared end style end fraction end root end style over denominator begin display style square root of 9 plus 5 over infinity squared end root minus square root of 4 minus fraction numerator begin display style 6 end style over denominator begin display style infinity squared end style end fraction end root end style end fraction right parenthesis end cell row blank equals cell fraction numerator begin display style square root of 1 minus 0 end root plus square root of 1 plus 0 end root end style over denominator begin display style square root of 9 plus 0 end root minus square root of 4 minus 0 end root end style end fraction right parenthesis end cell row blank equals cell fraction numerator begin display style 1 plus 1 end style over denominator begin display style 3 minus 2 end style end fraction end cell row blank equals 2 end table end style   

Jadi, Nilai dari Error converting from MathML to accessible text..

0

Roboguru

Tentukan nilai: x→∞lim​(2x−2−4x2−4x+3​)

Pembahasan Soal:

Ingat kembali :

begin mathsize 14px style limit as x rightwards arrow infinity of open parentheses square root of a x squared plus b x plus c end root minus square root of p x squared plus q x plus r end root close parentheses equals L end style 

  • begin mathsize 14px style L equals negative infinity end style jika dan hanya jika begin mathsize 14px style a less than p end style
  • L equals fraction numerator b minus q over denominator 2 square root of a end fraction jika dan hanya jika begin mathsize 14px style a equals p end style
  • begin mathsize 14px style L equals infinity end style jika dan hanya jika begin mathsize 14px style a greater than p end style

 

sehingga 

begin mathsize 12px style limit as x rightwards arrow infinity of left parenthesis 2 straight x minus 2 minus square root of 4 straight x squared minus 4 straight x plus 3 end root right parenthesis equals limit as x rightwards arrow infinity of open parentheses square root of left parenthesis 2 x minus 2 right parenthesis squared end root minus square root of 4 straight x squared minus 4 straight x plus 3 end root close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as straight x rightwards arrow infinity of open parentheses square root of 4 straight x squared minus 8 straight x plus 4 end root minus square root of 4 straight x squared minus 4 straight x plus 3 end root close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals L end style

karna nilai a equals p equals 4 maka 

L equals fraction numerator b minus q over denominator 2 square root of a end fraction space space space equals fraction numerator negative 8 minus left parenthesis negative 4 right parenthesis over denominator 2 square root of 4 end fraction space space space equals fraction numerator negative 8 plus 4 over denominator 2 times 2 end fraction equals fraction numerator negative 4 over denominator 4 end fraction equals negative 1

 

Jadi, nilai dari limit as x rightwards arrow infinity of left parenthesis 2 straight x minus 2 minus square root of 4 straight x squared minus 4 straight x plus 3 end root right parenthesis equals negative 1 .

0

Roboguru

Nilai x→∞lim​x5x+4​−3x+9​​=...

Pembahasan Soal:

Bagi pembilang dan penyebut dengan variabel pangkat tertinggi dari penyebut yaitu x.

Maka,

limx(4x5x+43x+9)==limxx4xx25x+x24x23x+x29limx4x5+x24x3+x29

Ingat bahwa

limxxnc=0 

maka,

limx(4x5x+43x+9)=====limx4x5+x24x3+x2940+00+0400400 

Jadi, nilai limit dari xlimx5x+43x+9 adalah 0.

Oleh karena itu, jawaban yang benar adalah A.

2

Roboguru

x→∞lim​25x2−5x+10​−(5x−3)=...

Pembahasan Soal:

Cara I

 

Ingat bentuk khusus berikut:

limit as x rightwards arrow infinity of square root of a x squared plus b x plus c end root minus square root of p x squared plus q x plus r end root equals L Jika space a equals p comma space maka space L equals fraction numerator b minus q over denominator 2 square root of a end fraction

Pertama ubah soal ke bentuk khusus,

limit as x rightwards arrow infinity of square root of 25 x squared minus 5 x plus 10 end root minus open parentheses 5 x minus 3 close parentheses equals limit as x rightwards arrow infinity of square root of 25 x squared minus 5 x plus 10 end root minus square root of open parentheses 5 x minus 3 close parentheses squared end root equals limit as x rightwards arrow infinity of square root of 25 x squared minus 5 x plus 10 end root minus square root of 25 x squared minus 30 x plus 9 end root

diperoleh

a equals p equals 25 b equals negative 5 q equals negative 30

Sehingga hasil limit adalah

table attributes columnalign right center left columnspacing 0px end attributes row L equals cell fraction numerator b minus q over denominator 2 square root of a end fraction end cell row blank equals cell fraction numerator negative 5 minus left parenthesis negative 30 right parenthesis over denominator 2 square root of 25 end fraction end cell row blank equals cell 25 over 10 end cell row blank equals cell 5 over 2 end cell end table

  Cara I

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow infinity of square root of 25 x squared minus 5 x plus 10 end root minus open parentheses 5 x minus 3 close parentheses end cell row blank equals cell limit as x rightwards arrow infinity of square root of 25 x squared minus 5 x plus 10 end root minus square root of open parentheses 5 x minus 3 close parentheses squared end root end cell row blank equals cell limit as x rightwards arrow infinity of square root of 25 x squared minus 5 x plus 10 end root minus square root of 25 x squared minus 30 x plus 9 end root end cell row blank equals cell limit as x rightwards arrow infinity of square root of 25 x squared minus 5 x plus 10 end root minus square root of 25 x squared minus 30 x plus 9 end root cross times fraction numerator square root of 25 x squared minus 5 x plus 10 end root plus square root of 25 x squared minus 30 x plus 9 end root over denominator square root of 25 x squared minus 5 x plus 10 end root plus square root of 25 x squared minus 30 x plus 9 end root end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator left parenthesis 25 x squared minus 5 x plus 10 right parenthesis minus left parenthesis 25 x squared minus 30 x plus 9 right parenthesis over denominator square root of 25 x squared minus 5 x plus 10 end root plus square root of 25 x squared minus 30 x plus 9 end root end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator 25 x plus 1 over denominator square root of 25 x squared minus 5 x plus 10 end root plus square root of 25 x squared minus 30 x plus 9 end root end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator 25 x over denominator x end fraction end style plus begin display style 1 over x end style over denominator square root of begin display style fraction numerator 25 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 10 over x squared end style end root plus square root of begin display style fraction numerator 25 x squared over denominator x squared end fraction end style minus begin display style fraction numerator 30 x over denominator x squared end fraction end style plus begin display style 9 over x squared end style end root end fraction end cell row blank blank blank row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style 25 plus 1 over x end style over denominator square root of begin display style 25 minus 5 over x plus 10 over x squared end style end root plus square root of begin display style 25 minus 30 over x plus 9 over x squared end style end root end fraction end cell row blank equals cell fraction numerator begin display style 25 plus 0 end style over denominator square root of begin display style 25 minus 0 plus 0 end style end root plus square root of begin display style 25 minus 0 plus 0 end style end root end fraction end cell row blank equals cell fraction numerator begin display style 25 end style over denominator 5 plus 5 end fraction end cell row blank equals cell 25 over 10 end cell row blank equals cell 5 over 2 end cell row blank blank blank end table end style

 

Jadi, limit as x rightwards arrow infinity of square root of 25 x squared minus 5 x plus 10 end root minus open parentheses 5 x minus 3 close parentheses equals 5 over 2.

0

Roboguru

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