Jika f(x)=3x−1, tentukan nilai h→0limhf(x+h)−f(x).

Pertanyaan

Jika $\style{font-size:14px}{f\left(x\right)=3x-1}$ , tentukan nilai

$begin mathsize 14px style limit as h rightwards arrow 0 of fraction numerator f open parentheses x plus h close parentheses minus f open parentheses x close parentheses over denominator h end fraction end style$ .

D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Jawaban

nilai dari $limit as h rightwards arrow 0 of fraction numerator f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis over denominator h end fraction$ adalah

Pembahasan

Diketahui f open parentheses x close parentheses equals 3 x minus 1 , maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x plus h close parentheses end cell equals cell 3 open parentheses x plus h close parentheses minus 1 end cell row blank equals cell 3 x plus 3 h minus 1 end cell end table end style

Karena jika disubstitusikan secara langsung menghasilkan bentuk tak tentu karena penyebutnya 0, maka selesaikan limit tersebut menggunakan metode pemfaktoran seperti berikut.

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as h rightwards arrow 0 of fraction numerator f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis over denominator h end fraction end cell equals cell limit as h rightwards arrow 0 of fraction numerator open parentheses 3 x plus 3 h minus 1 close parentheses minus open parentheses 3 x minus 1 close parentheses over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of fraction numerator 3 h over denominator h end fraction end cell row blank equals cell limit as h rightwards arrow 0 of 3 end cell row blank equals 3 end table end style$

Jadi, nilai dari $limit as h rightwards arrow 0 of fraction numerator f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis over denominator h end fraction$ adalah