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Nilai dari

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Nilai dari begin mathsize 14px style limit as straight x rightwards arrow infinity of fraction numerator 4 straight x to the power of 5 plus straight x squared minus 2 straight x minus 1 over denominator straight x squared plus 1 end fraction equals.... end style 

Pembahasan Soal:

Konsep limit  tak hingga: metode membagi dengan pangkat tertinggi

Diketahui: begin mathsize 14px style limit as straight x rightwards arrow infinity of fraction numerator 4 straight x to the power of 5 plus straight x squared minus 2 straight x minus 1 over denominator straight x squared plus 1 end fraction equals.... end style

Pangkat tertinggi adalah begin mathsize 14px style straight x to the power of 5 end style, maka pembilang dan penyebut dibagi oleh begin mathsize 14px style straight x to the power of 5 end style, maka diperoleh penyelesaiannya sebagai berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as straight x rightwards arrow infinity of fraction numerator 4 straight x to the power of 5 plus straight x squared minus 2 straight x minus 1 over denominator straight x squared plus 1 end fraction end cell equals cell limit as straight x rightwards arrow infinity of fraction numerator begin display style fraction numerator 4 straight x to the power of 5 over denominator straight x to the power of 5 end fraction end style plus begin display style straight x squared over straight x to the power of 5 end style minus begin display style fraction numerator 2 straight x over denominator straight x to the power of 5 end fraction end style minus begin display style 1 over straight x to the power of 5 end style over denominator begin display style straight x squared over straight x to the power of 5 end style plus begin display style 1 over straight x to the power of 5 end style end fraction end cell row blank equals cell limit as straight x rightwards arrow infinity of fraction numerator begin display style 4 plus 1 over straight x cubed minus 2 over straight x to the power of 4 minus 1 over straight x to the power of 5 end style over denominator begin display style 1 over straight x cubed plus 1 over straight x to the power of 5 end style end fraction end cell row blank equals cell fraction numerator begin display style 4 plus 1 over infinity cubed minus 2 over infinity to the power of 4 minus 1 over infinity to the power of 5 end style over denominator begin display style 1 over infinity cubed plus 1 over infinity to the power of 5 end style end fraction end cell row blank equals cell fraction numerator 4 plus 0 minus 0 minus 0 over denominator 0 plus 0 end fraction end cell row blank equals cell 4 over 0 end cell row blank equals infinity end table end style 

Jadi, Nilai begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as straight 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 4 straight x to the power of 5 plus straight x squared minus 2 straight x minus 1 over denominator straight x squared plus 1 end fraction end cell 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 end style.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

R. RGFLLIMA

Terakhir diupdate 30 April 2021

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Pertanyaan yang serupa

Hasil dari  adalah....

Pembahasan Soal:

Dengan menggunakan konsep limit menuju tak hingga diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of open parentheses 3 plus fraction numerator 3 over denominator 3 plus begin display style fraction numerator 3 over denominator 3 plus begin display style 3 over x end style end fraction end style end fraction close parentheses end cell equals cell limit as x rightwards arrow infinity of open parentheses 3 plus fraction numerator 3 over denominator 3 plus begin display style fraction numerator 3 over denominator begin display style fraction numerator 3 x plus 3 over denominator x end fraction end style end fraction end style end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow infinity of open parentheses 3 plus fraction numerator 3 over denominator 3 plus begin display style fraction numerator 3 x over denominator begin display style 3 x plus 3 end style end fraction end style end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow infinity of open parentheses 3 plus fraction numerator 3 over denominator begin display style fraction numerator 9 x plus 9 plus 3 x over denominator begin display style 3 x plus 3 end style end fraction end style end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow infinity of open parentheses 3 plus fraction numerator 3 over denominator begin display style fraction numerator 12 x plus 9 over denominator begin display style 3 x plus 3 end style end fraction end style end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow infinity of open parentheses 3 plus fraction numerator 3 open parentheses 3 x plus 3 close parentheses over denominator begin display style 12 x plus 9 end style end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow infinity of open parentheses 3 plus fraction numerator 9 x plus 9 over denominator begin display style 12 x plus 9 end style end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow infinity of open parentheses 3 plus fraction numerator 9 x plus 9 over denominator begin display style 12 x plus 9 end style end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow infinity of open parentheses fraction numerator 36 x plus 27 plus 9 x plus 9 over denominator begin display style 12 x plus 9 end style end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow infinity of open parentheses fraction numerator 45 x plus 36 over denominator begin display style 12 x plus 9 end style end fraction close parentheses 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 x rightwards arrow infinity of open parentheses fraction numerator begin display style fraction numerator 45 x over denominator x end fraction end style plus begin display style 36 over x end style over denominator begin display style fraction numerator begin display style 12 x end style over denominator x end fraction plus fraction numerator begin display style 9 end style over denominator x end fraction end style end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow infinity of open parentheses fraction numerator begin display style 45 plus 36 over x end style over denominator begin display style 12 plus fraction numerator begin display style 9 end style over denominator x end fraction end style end fraction close parentheses end cell end table 

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

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of open parentheses fraction numerator begin display style 45 plus 36 over x end style over denominator begin display style 12 plus fraction numerator begin display style 9 end style over denominator x end fraction end style end fraction close parentheses end cell equals cell fraction numerator 45 plus 0 over denominator 12 plus 0 end fraction end cell row blank equals cell 45 over 12 end cell row blank equals cell 15 over 4 end cell end table 

Dengan demikian hasil dari limit as x rightwards arrow infinity of open parentheses 3 plus fraction numerator 3 over denominator 3 plus begin display style fraction numerator 3 over denominator 3 plus begin display style 3 over x end style end fraction end style end fraction close parentheses equals 15 over 4.

0

Roboguru

Pembahasan Soal:

Gunakan konsep limit tak hingga. 

Perhatikan perhitungan berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell limit as x rightwards arrow infinity of space fraction numerator 4 x cubed plus x squared plus 1 over denominator 2 x to the power of 4 plus 3 end fraction end cell equals cell limit as x rightwards arrow infinity of space fraction numerator begin display style fraction numerator 4 x cubed over denominator x to the power of 4 end fraction end style plus begin display style x squared over x to the power of 4 end style plus begin display style 1 over x to the power of 4 end style over denominator begin display style fraction numerator 2 x to the power of 4 over denominator x to the power of 4 end fraction end style plus begin display style 3 over x to the power of 4 end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of space fraction numerator begin display style 4 over x end style plus begin display style 1 over x squared end style plus begin display style 1 over x to the power of 4 end style over denominator 2 plus begin display style 3 over x to the power of 4 end style end fraction end cell row blank equals cell fraction numerator 0 plus 0 plus 0 over denominator 2 plus 0 end fraction end cell row blank equals cell 0 over 2 end cell row cell limit as x rightwards arrow infinity of space fraction numerator 4 x cubed plus x squared plus 1 over denominator 2 x to the power of 4 plus 3 end fraction end cell equals 0 end table end style 

Jadi, diperoleh begin mathsize 14px style limit as x rightwards arrow infinity of space fraction numerator 4 x cubed plus x squared plus 1 over denominator 2 x to the power of 4 plus 3 end fraction equals 0 end style.

0

Roboguru

Nilai

Pembahasan Soal:

Langkah pertama untuk menentukan nilai limit limit as x rightwards arrow infinity of fraction numerator 3 x squared minus 2 x minus 6 over denominator 5 x minus 4 x squared end fraction adalah membagi setiap suku dengan variabel pangkat tertinggi. Variabel pangkat tertinggi pada bentuk limit tersebut adalah x squared, maka diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of fraction numerator 3 x squared minus 2 x minus 6 over denominator 5 x minus 4 x squared end fraction end cell equals cell limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator 3 x squared over denominator x squared end fraction end style minus begin display style fraction numerator 2 x over denominator x squared end fraction end style minus begin display style 6 over x squared end style over denominator begin display style fraction numerator 5 x over denominator x squared end fraction end style minus begin display style fraction numerator 4 x squared over denominator x squared end fraction end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator 3 minus begin display style 2 over x end style minus begin display style 6 over x squared end style over denominator begin display style 5 over x end style minus 4 end fraction end cell end table

Kemudian, dengan menggunakan teorema jika k bilangan asli maka limit as x rightwards arrow infinity of 1 over x to the power of k equals 0, diperoleh nilai limit tersebut sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of fraction numerator 3 x squared minus 2 x minus 6 over denominator 5 x minus 4 x squared end fraction end cell equals cell limit as x rightwards arrow infinity of fraction numerator 3 minus begin display style 2 over x end style minus begin display style 6 over x squared end style over denominator begin display style 5 over x end style minus 4 end fraction end cell row blank equals cell fraction numerator 3 minus 0 minus 0 over denominator 0 minus 4 end fraction end cell row blank equals cell fraction numerator 3 over denominator negative 4 end fraction end cell row blank equals cell negative 3 over 4 end cell end table


Dengan demikian, nilai limit as x rightwards arrow infinity of fraction numerator 3 x squared minus 2 x minus 6 over denominator 5 x minus 4 x squared end fraction equals negative 3 over 4.

0

Roboguru

Nilai dari

Pembahasan Soal:

Ingat bahwa, pada limit tak hingga apabila pangkat tertinggi pembilang sama dengan pangkat tertinggi penyebut, maka hasil limitnya adalah perbandingan koefisien pangkat tertinggi pembilang dan penyebut tersebut.

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of fraction numerator 4 straight x squared minus 5 straight x minus 3 over denominator 7 plus 3 x minus 6 straight x squared end fraction end cell equals cell limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator 4 straight x squared minus 5 straight x minus 3 over denominator x squared end fraction end style over denominator begin display style fraction numerator 7 plus 3 x minus 6 straight x squared over denominator x squared 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 4 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 minus begin display style 3 over x squared end style over denominator begin display style 7 over x squared end style plus begin display style fraction numerator 3 x over denominator x squared end fraction end style minus begin display style fraction numerator 6 x squared over denominator x squared end fraction end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style 4 minus 5 over x minus 3 over x squared end style over denominator begin display style 7 over x squared plus 3 over x minus 6 end style end fraction end cell row blank equals cell fraction numerator 4 minus 0 minus 0 over denominator 0 plus 0 minus 6 end fraction end cell row blank equals cell fraction numerator 4 over denominator negative 6 end fraction end cell row blank equals cell negative 2 over 3 end cell end table

Oleh karena itu, jawaban yang benar adalah D.

0

Roboguru

Nilai limit dari:

Pembahasan Soal:

Dengan metode pembagian pangkat tertinggi.

begin mathsize 14px style limit as straight x rightwards arrow infinity of fraction numerator straight x over denominator 1 plus straight x squared end fraction end style

dari fungsi aljabar di atas, pangkat tertinggi adalah begin mathsize 14px style x squared end style maka pembilang dan penyebut sama-sama dibagi begin mathsize 14px style x squared 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 straight x over denominator 1 plus straight x squared end fraction end cell equals cell limit as straight x rightwards arrow infinity of fraction numerator begin display style straight x over straight x squared end style over denominator begin display style 1 over straight x squared end style plus begin display style straight x squared over straight x squared end style end fraction end cell row blank equals cell limit as straight x rightwards arrow infinity of fraction numerator begin display style 1 over straight x end style over denominator begin display style 1 over straight x squared plus 1 end style end fraction end cell row blank equals cell fraction numerator 0 over denominator 0 plus 1 end fraction end cell row blank equals 0 end table 

Jadi, nilai dari begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as straight 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 straight x over denominator 1 plus straight x squared end fraction end cell 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 0 end table end style .

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