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Nilai dari  adalah ....

Pertanyaan

Nilai dari stack l i m with x rightwards arrow infinity below fraction numerator 4 x minus 7 over denominator 3 x plus 5 end fraction adalah ....

Pembahasan Soal:

Ingat bahwa, pada limit tak hingga pembagian dua fungsi, maka nilai limitnya merupakan perbandingan koefisien pangkat tertinggi pembilang dengan penyebutnya. Sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell stack l i m with x rightwards arrow infinity below fraction numerator 4 x minus 7 over denominator 3 x plus 5 end fraction end cell equals cell fraction numerator begin display style fraction numerator 4 x minus 7 over denominator x end fraction end style over denominator begin display style fraction numerator 3 x plus 5 over denominator x end fraction end style end fraction end cell row blank equals cell fraction numerator 4 minus 0 over denominator 3 plus 0 end fraction end cell row blank equals cell 4 over 3 end cell end table

Oleh karena itu, nilai dari stack l i m with x rightwards arrow infinity below fraction numerator 4 x minus 7 over denominator 3 x plus 5 end fraction adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 4 over 3 end cell end table.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

F. Kurnia

Mahasiswa/Alumni Universitas Jember

Terakhir diupdate 06 Juni 2021

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Pertanyaan yang serupa

Nilai dari  =

Pembahasan Soal:

Dengan menggunakan konsep limit tak hingga

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limit as x rightwards arrow infinity of fraction numerator 2 x cubed plus 3 x squared plus 7 over denominator x squared plus 3 x plus 4 end fractionmempunyai pangkat tertinggi pada pembilangnya 3 dan pangkat tertinggi pada penyebutnya 2, sehingga berdasarkan rumus diatas m greater than n maka :

 

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of fraction numerator 2 x cubed plus 3 x squared plus 7 over denominator x squared plus 3 x plus 4 end fraction end cell equals infinity row blank blank blank end table 

Jadi, nilaibegin mathsize 14px style limit as x rightwards arrow infinity of fraction numerator 2 x cubed plus 3 x squared plus 7 over denominator x squared plus 3 x plus 4 end fraction end style  adalah begin mathsize 14px style infinity end style.

Roboguru

Nilai limit  untuk t mendekati tak hingga adalah ....

Pembahasan Soal:

Perhatikan bentuk umum limit tak hingga berikut :

limit as x rightwards arrow infinity of fraction numerator a x to the power of m plus b x to the power of m minus 1 end exponent plus horizontal ellipsis plus c over denominator p x to the power of n plus q x to the power of n minus 1 end exponent plus horizontal ellipsis plus r end fraction.

Penyelesaian limit di atas adalah :

jika m greater than n maka L equals infinity,
jika m equals n maka L equals a over p, dan
jika m less than n maka L equals 0.

Pada soal diberikan limit as t rightwards arrow infinity of space fraction numerator t squared minus 4 over denominator t minus 2 end fraction.

Perhatikan bahwa pangkat tertinggi pada pembilang left parenthesis m equals 2 right parenthesis lebih besar nilainya daripada pangkat tertinggi pada penyebut left parenthesis n equals 1 right parenthesis, atau m greater than n, sehingga

limit as t rightwards arrow infinity of space fraction numerator t squared minus 4 over denominator t minus 2 end fraction equals infinity.

Jadi, nilai limit fungsi f left parenthesis t right parenthesis untuk t mendekati tak hingga adalah infinity.

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

Roboguru

Tentukan nilai limit fungsi berikut.

Pembahasan Soal:

Dengan menggunakan aturan limit untuk pembagian 2 buah fungsi, maka nilai limit as x rightwards arrow infinity of space fraction numerator 2 x squared plus 3 over denominator 4 x squared plus 2 x end fraction adalah

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

Dengan demikian, nilai dari limit as x rightwards arrow infinity of space fraction numerator 2 x squared plus 3 over denominator 4 x squared plus 2 x end fraction adalah 1 half. 

Roboguru

Hitunglah nilai limit dari fungsi-fungsi berikut ini! e.

Pembahasan Soal:

Nilai limit dengan bentuk umum :

 begin mathsize 14px style limit as x rightwards arrow infinity of fraction numerator a subscript 1 x to the power of m plus a subscript 2 x to the power of m minus 1 end exponent plus a subscript 3 m to the power of m minus 2 end exponent plus space... space plus a subscript m over denominator b subscript 1 x to the power of n plus b subscript 2 x to the power of n minus 1 end exponent plus b subscript 3 x to the power of n minus 2 end exponent plus space... space plus b subscript n end fraction end style adalah begin mathsize 14px style open curly brackets table attributes columnalign left end attributes row cell 0 space untuk space m less than n end cell row cell infinity space untuk space m greater than n end cell row cell a subscript 1 over b subscript 1 untuk space m equals n end cell end table close end style.

Karena begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell limit as x rightwards arrow infinity of fraction numerator open parentheses 2 x plus 1 close parentheses squared over denominator x minus 3 end fraction end cell equals cell limit as x rightwards arrow infinity of fraction numerator 2 x squared plus 4 x plus 1 over denominator x minus 3 end fraction end cell end table end style, maka undefined dan berlaku undefined sehingga dapat diperoleh:

begin mathsize 14px style limit as x rightwards arrow infinity of fraction numerator open parentheses 2 x plus 1 close parentheses squared over denominator x minus 3 end fraction equals infinity end style

Dengan demikian nilai dari begin mathsize 14px style limit as x rightwards arrow infinity of fraction numerator open parentheses 2 x plus 1 close parentheses squared over denominator x minus 3 end fraction end style adalah undefined.

Roboguru

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