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Nilai dari stack l i m with x rightwards arrow infinity below open parentheses square root of 4 x squared plus 8 x plus 3 end root minus square root of 4 x open parentheses x plus 2 close parentheses end root close parentheses equals....

  1. negative 8

  2. negative 4

  3. 0

  4. 4

  5. 8

Pembahasan Video:

Pembahasan Soal:

Limit x → ∞ dari suatu fungsi bentuk akar dapat diselesaikan dengan menggunakan rumus:

xLimax2+bx+cpx2+qx+r=2abq,untuka=p

jadi, nilai limitnya yaitu


xlim(4x2+8x+34x(x+2))====xlim(4x2+8x+34x2+8x)2488400

Jadi, jawaban yang tepat adalah C.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

N. Puspita

Terakhir diupdate 14 September 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Tentukan nilai:

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.

0

Roboguru

Tentukan nilai:

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

Tentukan nilai:

Pembahasan Soal:

Dengan menerapkan konsep perkalian sekawan:

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 left parenthesis square root of straight x plus 6 square root of straight x end root minus square root of straight x minus 2 square root of straight x end root right parenthesis end cell row blank equals cell limit as straight x rightwards arrow infinity of left parenthesis square root of straight x plus 6 square root of straight x end root minus square root of straight x minus 2 square root of straight x end root right parenthesis times fraction numerator left parenthesis square root of straight x plus 6 square root of straight x end root plus square root of straight x minus 2 square root of straight x end root right parenthesis over denominator left parenthesis square root of straight x plus 6 square root of straight x end root plus square root of straight x minus 2 square root of straight x end root right parenthesis end fraction end cell row blank equals cell limit as straight x rightwards arrow infinity of fraction numerator straight x plus 6 square root of straight x minus left parenthesis straight x minus 2 square root of straight x right parenthesis over denominator left parenthesis square root of straight x plus 6 square root of straight x end root plus square root of straight x minus 2 square root of straight x end root right parenthesis end fraction end cell row blank equals cell limit as straight x rightwards arrow infinity of fraction numerator straight x plus 6 square root of straight x minus straight x plus 2 square root of straight x over denominator left parenthesis square root of straight x plus 6 square root of straight x end root plus square root of straight x minus 2 square root of straight x end root right parenthesis end fraction end cell row blank equals cell limit as straight x rightwards arrow infinity of fraction numerator 8 square root of straight x over denominator left parenthesis square root of straight x plus 6 square root of straight x end root plus square root of straight x minus 2 square root of straight x end root right parenthesis end fraction end cell end table end style  

Pangkat undefined terbesar dari fungsi table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 8 square root of straight x over denominator left parenthesis square root of straight x plus 6 square root of straight x end root plus square root of straight x minus 2 square root of straight x end root right parenthesis end fraction end cell end table adalah begin mathsize 14px style 1 half end style atau begin mathsize 14px style square root of x end style maka pembilang dan penyebut di bagi oleh begin mathsize 14px style square root of x end style, sehingga diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as straight x rightwards arrow infinity of fraction numerator 8 square root of straight x over denominator left parenthesis square root of straight x plus 6 square root of straight x end root plus square root of straight x minus 2 square root of straight x end root right parenthesis end fraction end cell equals cell limit as straight x rightwards arrow infinity of fraction numerator begin display style fraction numerator 8 square root of straight x over denominator square root of straight x end fraction end style over denominator begin display style fraction numerator left parenthesis square root of straight x plus 6 square root of straight x end root plus square root of straight x minus 2 square root of straight x end root right parenthesis over denominator square root of straight x end fraction end style end fraction end cell row blank equals cell limit as straight x rightwards arrow infinity of fraction numerator begin display style 8 end style over denominator begin display style fraction numerator begin display style left parenthesis square root of straight x plus 6 square root of straight x end root end style over denominator square root of straight x end fraction plus fraction numerator begin display style square root of straight x minus 2 square root of straight x end root right parenthesis end style over denominator square root of straight x end fraction end style end fraction end cell row blank equals cell limit as straight x rightwards arrow infinity of fraction numerator begin display style 8 end style over denominator begin display style square root of fraction numerator straight x plus 6 square root of straight x over denominator straight x end fraction end root plus square root of fraction numerator straight x minus 2 square root of straight x over denominator straight x end fraction end root end style end fraction end cell row blank equals cell limit as straight x rightwards arrow infinity of fraction numerator begin display style 8 end style over denominator begin display style square root of straight x over straight x plus fraction numerator 6 square root of straight x over denominator straight x end fraction end root plus square root of straight x over straight x minus fraction numerator 2 square root of straight x over denominator straight x end fraction end root end style end fraction end cell row blank equals cell limit as straight x rightwards arrow infinity of fraction numerator begin display style 8 end style over denominator begin display style square root of 1 plus fraction numerator 6 over denominator square root of straight x end fraction end root plus square root of 1 minus fraction numerator 2 over denominator square root of straight x end fraction end root end style end fraction end cell row blank equals cell fraction numerator begin display style 8 end style over denominator begin display style square root of 1 plus fraction numerator 6 over denominator square root of infinity end fraction end root plus square root of 1 minus fraction numerator 2 over denominator square root of infinity end fraction end root end style end fraction end cell row blank equals cell fraction numerator begin display style 8 end style over denominator begin display style square root of 1 plus 0 end root plus square root of 1 minus 0 end root end style end fraction end cell row blank equals cell 8 over 2 end cell row blank equals 4 end table 

Jadi, Nilai limit as straight X rightwards arrow infinity of left parenthesis square root of straight x plus 6 square root of straight x end root minus square root of straight x minus 2 square root of straight x end root right parenthesis equals 4 

 

 

0

Roboguru

Pembahasan Soal:

Pada limit tak hingga, misal


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 L equals fraction numerator b minus q over denominator 2 square root of a end fraction comma space j i k a space a equals p

maka, nilai limitnya adalah

begin mathsize 14px 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 open parentheses 2 x plus 1 close parentheses squared end root minus square root of 4 x squared plus 3 x plus 9 end root end cell row blank equals cell limit as x rightwards arrow infinity of square root of 4 x squared plus 4 x plus 1 end root minus square root of 4 x squared plus 3 x plus 9 end root end cell row blank equals cell fraction numerator b minus q over denominator 2 square root of a end fraction end cell row blank equals cell fraction numerator 4 minus 3 over denominator 2 square root of 4 end fraction end cell row blank equals cell 1 fourth end cell end table end style

1

Roboguru

Hitunglah nilai dari .

Pembahasan Soal:

limit as x rightwards arrow infinity of open parentheses square root of 4 x squared plus x end root plus square root of 9 x squared minus x end root minus 5 x close parentheses equals limit as x rightwards arrow infinity of open parentheses square root of 4 x squared plus x end root plus square root of 9 x squared minus x end root minus open parentheses 2 x plus 3 x close parentheses close parentheses limit as x rightwards arrow infinity of open parentheses square root of 4 x squared plus x end root plus square root of 9 x squared minus x end root minus 5 x close parentheses equals limit as x rightwards arrow infinity of open parentheses square root of 4 x squared plus x end root plus square root of 9 x squared minus x end root minus 2 x minus 3 x close parentheses limit as x rightwards arrow infinity of open parentheses square root of 4 x squared plus x end root plus square root of 9 x squared minus x end root minus 5 x close parentheses equals limit as x rightwards arrow infinity of open parentheses square root of 4 x squared plus x end root minus 2 x plus square root of 9 x squared minus x end root minus 3 x close parentheses limit as x rightwards arrow infinity of open parentheses square root of 4 x squared plus x end root plus square root of 9 x squared minus x end root minus 5 x close parentheses equals limit as x rightwards arrow infinity of open parentheses square root of 4 x squared plus x end root minus square root of 4 x squared end root plus square root of 9 x squared minus x end root minus square root of 9 x squared end root close parentheses limit as x rightwards arrow infinity of open parentheses square root of 4 x squared plus x end root plus square root of 9 x squared minus x end root minus 5 x close parentheses equals limit as x rightwards arrow infinity of open parentheses square root of 4 x squared plus x end root minus square root of 4 x squared end root close parentheses plus limit as x rightwards arrow infinity of open parentheses square root of 9 x squared minus x end root minus square root of 9 x squared end root close parentheses text Bentuk diatas sama dengan end text space limit as x rightwards arrow infinity of left parenthesis 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 space text dan karena end text space space a equals p space text maka nilai limitnya adalah end text space fraction numerator b minus q over denominator 2 square root of a end fraction comma space text sehingga end text limit as x rightwards arrow infinity of open parentheses square root of 4 x squared plus x end root plus square root of 9 x squared minus x end root minus 5 x close parentheses equals fraction numerator 1 minus 0 over denominator 2 square root of 4 end fraction plus fraction numerator negative 1 minus 0 over denominator 2 square root of 9 end fraction limit as x rightwards arrow infinity of open parentheses square root of 4 x squared plus x end root plus square root of 9 x squared minus x end root minus 5 x close parentheses equals 1 fourth minus 1 over 6 limit as x rightwards arrow infinity of open parentheses square root of 4 x squared plus x end root plus square root of 9 x squared minus x end root minus 5 x close parentheses equals fraction numerator 3 minus 2 over denominator 12 end fraction equals 1 over 12

Jadi, hasil limitnya adalah 1 over 12.

0

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