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Nilai dari x→∞lim​5x​+7x2−2x3(3x2+x​)(x−3x​)​ adalah ...

Pertanyaan

Nilai dari limit as x rightwards arrow infinity of fraction numerator open parentheses 3 x squared plus square root of x close parentheses open parentheses x minus 3 square root of x close parentheses over denominator 5 square root of x plus 7 x squared minus 2 x cubed end fraction adalah ...
 

  1. 3 over 2

  2. 0

  3. infinity

  4. negative 3 over 2

  5. negative 2 over 3

Pembahasan Video:

Pembahasan Soal:

Untuk menyelesaikan limit menuju tak hingga open parentheses x rightwards arrow infinity close parentheses, kita gunakan limit dasarnya yaitu,

limit as x rightwards arrow infinity of a over x to the power of n equals 0

dengan a bilangan real dan n bilangan asli, artinya kita harus mengarahkan bentuk limit tak hingga menjadi rumus dasar dengan cara :

  1. Buat fungsinya menjadi bentuk pecahan, jika bentuknya dalam akar mak akalikan dengan bentuk sekawannya (merasionalkan)
  2. Bagi variabelnya dengan pangkat tertinggi

Maka,

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of fraction numerator open parentheses 3 x squared plus square root of x close parentheses open parentheses x minus 3 square root of x close parentheses over denominator 5 square root of x plus 7 x squared minus 2 x cubed end fraction end cell equals cell limit as x rightwards arrow infinity of fraction numerator 3 x cubed minus 9 x squared square root of x plus x square root of x minus 3 square root of x squared end root over denominator 5 square root of x plus 7 x squared minus 2 x cubed end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator 3 x cubed over denominator x cubed end fraction end style minus begin display style fraction numerator 9 x squared square root of x over denominator x cubed end fraction end style plus begin display style fraction numerator x square root of x over denominator x cubed end fraction end style minus begin display style fraction numerator 3 square root of x squared end root over denominator x cubed end fraction end style over denominator begin display style fraction numerator 5 square root of x over denominator x cubed end fraction end style plus begin display style fraction numerator 7 x squared over denominator x cubed end fraction end style minus begin display style fraction numerator 2 x cubed over denominator x cubed end fraction end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator 3 x cubed over denominator x cubed end fraction minus fraction numerator 9 x squared square root of x over denominator x cubed end fraction plus fraction numerator x square root of x over denominator x cubed end fraction minus fraction numerator 3 square root of x squared end root over denominator x cubed end fraction end style over denominator begin display style fraction numerator 5 square root of x over denominator x cubed end fraction plus fraction numerator 7 x squared over denominator x cubed end fraction minus fraction numerator 2 x cubed over denominator x cubed end fraction end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style 3 minus fraction numerator 9 x squared times x to the power of begin display style 1 half end style end exponent over denominator x cubed end fraction plus fraction numerator x times x to the power of 1 half end exponent over denominator x cubed end fraction minus fraction numerator 3 x over denominator x cubed end fraction end style over denominator begin display style fraction numerator 5 times x to the power of 1 half end exponent over denominator x cubed end fraction plus 7 over x minus 2 end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style 3 minus fraction numerator 9 x to the power of 2 plus begin display style 1 half end style end exponent over denominator x cubed end fraction plus x to the power of 1 plus begin display style 1 half end style end exponent over x cubed minus 3 over x squared end style over denominator begin display style fraction numerator 5 x to the power of 1 half end exponent over denominator x cubed end fraction plus 7 over x minus 2 end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style 3 minus fraction numerator 9 x to the power of begin display style 5 over 2 end style end exponent over denominator x cubed end fraction plus x to the power of begin display style 3 over 2 end style end exponent over x cubed minus 3 over x squared end style over denominator begin display style fraction numerator 5 x to the power of 1 half end exponent over denominator x cubed end fraction plus 7 over x minus 2 end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style 3 minus 9 over x to the power of begin display style 1 half end style end exponent plus x over x to the power of begin display style 3 over 2 end style end exponent minus 3 over x squared end style over denominator begin display style 5 over x to the power of begin display style 5 over 2 end style end exponent plus 7 over x minus 2 end style end fraction end cell row blank equals cell fraction numerator begin display style 3 minus 9 over infinity to the power of begin display style 1 half end style end exponent plus x over infinity to the power of begin display style 3 over 2 end style end exponent minus 3 over infinity squared end style over denominator begin display style 5 over infinity to the power of begin display style 1 half end style end exponent plus 7 over infinity minus 2 end style end fraction end cell row blank equals cell fraction numerator begin display style 3 minus 0 plus 0 minus 0 end style over denominator begin display style 0 plus 0 minus 2 end style end fraction end cell row blank equals cell fraction numerator 3 over denominator negative 2 end fraction end cell row blank equals cell negative 3 over 2 end cell end table end style


Jadi, nilai dari limit as x rightwards arrow infinity of fraction numerator open parentheses 3 x squared plus square root of x close parentheses open parentheses x minus 3 square root of x close parentheses over denominator 5 square root of x plus 7 x squared minus 2 x cubed end fraction adalah negative 3 over 2.


Oleh karena itu, jawaban yang benar adalah D.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

N. Puspita

Terakhir diupdate 07 Oktober 2021

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Pertanyaan yang serupa

Nilai dari x→∞lim​(4x2+8x+3​−4x(x+2)​)=....

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.

0

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.

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Roboguru

x→∞lim​2x−36x2+5x+1​​=...

Pembahasan Soal:

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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

x→∞lim​(16x2+9x−2​−(4x+1))

Pembahasan Soal:

Limittak hingga

Terdapat beberapa cara untuk menentukan limit menuju tak hingga dengan bentuk akar, seperti berikut ini:

limit as x rightwards arrow infinity of open parentheses square root of p x squared plus q x plus r end root minus square root of s x squared plus t x plus u end root close parentheses equals open curly brackets table attributes columnalign left end attributes row infinity row cell fraction numerator q minus t over denominator 2 square root of p end fraction end cell row cell negative infinity end cell end table close  

Diperoleh begin mathsize 14px style infinity end style apabila p greater than s, diperoleh fraction numerator q minus t over denominator 2 square root of p end fraction untuk p equals s, dan begin mathsize 14px style negative infinity end style untuk p less than s.

 

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of open parentheses square root of 16 x squared plus 9 x minus 2 end root minus open parentheses 4 x plus 1 close parentheses close parentheses end cell equals cell limit as x rightwards arrow infinity of open parentheses square root of 16 x squared plus 9 x minus 2 end root minus square root of 16 x squared plus 8 x plus 1 end root close parentheses end cell row blank equals cell fraction numerator 9 minus 8 over denominator 2 square root of 16 end fraction end cell row blank equals cell fraction numerator 1 over denominator 2 times 4 end fraction end cell row blank equals cell 1 over 8 end cell end table end style

Jadi, limit as x rightwards arrow infinity of open parentheses square root of 16 x squared plus 9 x minus 2 end root minus open parentheses 4 x plus 1 close parentheses close parenthesesequals 1 over 8.

0

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