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Pertanyaan

limit as x rightwards arrow infinity of open parentheses 4 x minus 3 close parentheses squared over open parentheses 2 x plus 5 close parentheses squared equals.... 

Pembahasan Soal:

Ingat bahwa

limit as x rightwards arrow infinity of 1 over x to the power of n equals 0 comma space n greater than 0

Maka dapat diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of open parentheses 4 x minus 3 close parentheses squared over open parentheses 2 x plus 5 close parentheses squared end cell equals cell limit as x rightwards arrow infinity of fraction numerator 16 x squared minus 24 x plus 9 over denominator 4 x squared plus 20 x plus 25 end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator x squared open parentheses 16 minus begin display style 24 over x end style plus begin display style 9 over x squared end style close parentheses over denominator x squared open parentheses 4 plus begin display style 20 over x end style plus begin display style 25 over x squared end style close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator 16 minus begin display style 24 over x end style plus begin display style 9 over x squared end style over denominator 4 plus begin display style 20 over x end style plus begin display style 25 over x squared end style end fraction end cell row blank equals cell fraction numerator 16 minus 0 plus 0 over denominator 4 plus 0 plus 0 end fraction end cell row blank equals 4 end table

Jadi, limit as x rightwards arrow infinity of open parentheses 4 x minus 3 close parentheses squared over open parentheses 2 x plus 5 close parentheses squared equals 4 

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

S. Lestari

Terakhir diupdate 30 Agustus 2021

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Pertanyaan yang serupa

Nilai dari   adalah ....

Pembahasan Soal:

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

Maka diperoleh perhitungan:

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of fraction numerator x squared plus 2 x minus 2021 over denominator 3 x squared plus 2021 end fraction end cell equals cell limit as x rightwards arrow infinity of fraction numerator x squared plus 2 x minus 2021 over denominator 3 x squared plus 2021 end fraction cross times fraction numerator begin display style 1 over x squared end style over denominator begin display style 1 over x squared end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style x squared over x squared end style plus begin display style fraction numerator 2 x over denominator x squared end fraction end style minus begin display style 2021 over x squared end style over denominator begin display style fraction numerator 3 x squared over denominator x squared end fraction end style plus begin display style 2021 over x squared end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style 1 plus 2 over x minus 2021 over x squared end style over denominator begin display style 3 plus 2021 over x squared end style end fraction end cell row blank equals cell fraction numerator begin display style 1 plus 2 over infinity minus 2021 over infinity squared end style over denominator begin display style 3 plus 2021 over infinity squared end style end fraction end cell row blank equals cell fraction numerator 1 plus 0 plus 0 over denominator 3 plus 0 end fraction end cell row blank equals cell 1 third end cell end table

Jadi table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow infinity of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator x squared plus 2 x minus 2021 over denominator 3 x squared plus 2021 end fraction end cell end table equals 1 third.

0

Roboguru

Nilai dari

Pembahasan Soal:

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 open parentheses 3 x squared minus 7 x plus 4 close parentheses cubed over open parentheses 2 x cubed minus 8 x plus 5 close parentheses squared end cell equals cell limit as x rightwards arrow infinity of fraction numerator open parentheses 3 x squared close parentheses cubed plus... plus 4 cubed over denominator open parentheses 2 x cubed close parentheses squared plus... plus 5 squared end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator 27 x to the power of 6 plus... plus 64 over denominator 4 x to the power of 6 plus... plus 25 end fraction cross times fraction numerator begin display style 1 over x to the power of 6 end style over denominator begin display style 1 over x to the power of 6 end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator 27 x to the power of 6 over denominator x to the power of 6 end fraction end style plus... plus begin display style 64 over x to the power of 6 end style over denominator begin display style fraction numerator 4 x to the power of 6 over denominator x to the power of 6 end fraction end style plus... plus begin display style 25 over x to the power of 6 end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style 27 plus... plus 64 over x to the power of 6 end style over denominator begin display style 4 plus... plus 25 over x to the power of 6 end style end fraction end cell row blank equals cell fraction numerator begin display style 27 plus... plus 0 end style over denominator begin display style 4 plus... plus 0 end style end fraction end cell row blank equals cell 27 over 4 end cell end table

Jadi, nilai limitnya adalah begin mathsize 14px style 27 over 4 end style.

0

Roboguru

Hitunglah limit fungsi berikut.

Pembahasan Soal:

Dalam hal ini begin mathsize 14px style f left parenthesis x right parenthesis end style dimanipulasi dengan cara:
Jika begin mathsize 14px style n end style adalah derajat tertinggi antara begin mathsize 14px style g left parenthesis x right parenthesis end style atau begin mathsize 14px style h left parenthesis x right parenthesis end style maka begin mathsize 14px style g left parenthesis x right parenthesis end style dan begin mathsize 14px style h left parenthesis x right parenthesis end style masing-masing dibagi dengan begin mathsize 14px style x to the power of n end style 
Memiliki pangkat tertinggi begin mathsize 14px style square root of x end style 

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of fraction numerator square root of x plus 1 end root over denominator square root of 4 x minus 1 end root end fraction end cell equals cell limit as x rightwards arrow infinity of fraction numerator square root of x over x plus 1 over x end root over denominator square root of fraction numerator 4 x over denominator x end fraction minus 1 over x end root end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator square root of 1 plus 1 over x end root over denominator square root of 4 minus 1 over x end root end fraction end cell row blank equals cell fraction numerator square root of 1 plus 0 end root over denominator square root of 4 minus 0 end root end fraction end cell row blank equals cell fraction numerator square root of 1 over denominator square root of 4 end fraction end cell row blank equals cell 1 half end cell end table

Dengan demikian, nilai dari begin mathsize 14px style limit as x rightwards arrow infinity of fraction numerator square root of x plus 1 end root over denominator square root of 4 x minus 1 end root end fraction equals 1 half end style.

0

Roboguru

Pembahasan Soal:

Untuk menentukan limit di atas dapat menggunakan konsep limit tak hinggga bentuk fungsi rasional, yaitu dengan membagi pembilang dan penyebut dengan variabel yang memiliki pangkat tertinggi di penyebut 

P e n y e b u t space p a d a space s o a l space equals space 6 x minus 3 comma space m a k a space v a r i a b e l space y a n g space m e m i l i k i space p a n g k a t space t e r t i n g g i space a d a l a h space x

Sehingga :

limit as x rightwards arrow infinity of fraction numerator 3 x squared plus 2 over denominator 6 x minus 3 end fraction times fraction numerator begin display style 1 over x end style over denominator begin display style 1 over x end style end fraction equals limit as x rightwards arrow infinity of fraction numerator 3 x plus begin display style 2 over x end style over denominator 6 minus begin display style 3 over x end style end fraction space space space space space space space space space space space space table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell bold lim with bold x bold rightwards arrow bold infinity below end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell bold C over bold x to the power of bold n end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank bold equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold 0 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold semicolon end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold italic n end table table attributes columnalign right center left columnspacing 0px end attributes row blank bold greater than blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold 0 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank bold space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table 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 table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow infinity of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 3 x over denominator begin display style 6 end style end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table space space space space space space space space space space space space space space space space space space space space space space space table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank infinity end table

Jadi, nilai limit di atas adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank infinity end table

0

Roboguru

Nilai dari  adalah ....

Pembahasan Soal:

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 3 straight x to the power of 4 minus 2 straight x squared plus straight x over denominator left parenthesis straight x plus 2 right parenthesis left parenthesis straight x minus 3 right parenthesis end fraction end cell equals cell limit as straight x rightwards arrow infinity of fraction numerator 3 straight x to the power of 4 minus 2 straight x squared plus straight x over denominator straight x squared minus straight x minus 6 end fraction end cell row blank equals cell limit as straight x rightwards arrow infinity of fraction numerator 3 straight x to the power of 4 minus 2 straight x squared plus straight x over denominator straight x squared minus straight x minus 6 end fraction cross times fraction numerator begin display style 1 over x to the power of 4 end style over denominator begin display style 1 over x to the power of 4 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 3 straight x to the power of 4 over denominator straight x to the power of 4 end fraction minus fraction numerator 2 straight x squared over denominator straight x to the power of 4 end fraction plus straight x over straight x to the power of 4 end style over denominator begin display style straight x squared over straight x to the power of 4 minus straight x over straight x to the power of 4 minus 6 over straight x to the power of 4 end style end fraction end cell row blank equals cell limit as straight x rightwards arrow infinity of fraction numerator begin display style 3 minus 2 over straight x squared plus 1 over straight x cubed end style over denominator begin display style 1 over straight x squared minus 1 over straight x cubed minus 6 over straight x to the power of 4 end style end fraction space end cell row blank equals cell fraction numerator begin display style 3 minus 2 over infinity squared plus 1 over infinity cubed end style over denominator begin display style 1 over infinity squared minus 1 over infinity cubed minus 6 over infinity to the power of 4 end style end fraction end cell row blank equals cell fraction numerator begin display style 3 minus 0 plus 0 end style over denominator begin display style 0 minus 0 minus 0 end style end fraction end cell row blank equals cell fraction numerator begin display style 3 end style over denominator begin display style 0 end style end fraction end cell row blank equals infinity end table  

Jadi, Nilai dari begin mathsize 14px style limit as straight x rightwards arrow infinity of fraction numerator 3 straight x to the power of 4 minus 2 straight x squared plus straight x over denominator left parenthesis straight x plus 2 right parenthesis left parenthesis straight x minus 3 right parenthesis end fraction end style adalah begin mathsize 14px style infinity end style.

0

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