Hitunglah nilai limit fungsi berikut: x→1limx−164(x+31−3x+52)

Pertanyaan

Hitunglah nilai limit fungsi berikut:

$begin mathsize 14px style limit as x rightwards arrow 1 of fraction numerator 64 over denominator x minus 1 end fraction open parentheses fraction numerator 1 over denominator x plus 3 end fraction minus fraction numerator 2 over denominator 3 x plus 5 end fraction close parentheses end style$

D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Pembahasan

Kita selesaikan limit tersebut dengan metode substitusi terlebih dahulu.

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 1 of fraction numerator 64 over denominator x minus 1 end fraction open parentheses fraction numerator 1 over denominator x plus 3 end fraction minus fraction numerator 2 over denominator 3 x plus 5 end fraction close parentheses end cell equals cell fraction numerator 64 over denominator 1 minus 1 end fraction open parentheses fraction numerator 1 over denominator 1 plus 3 end fraction minus fraction numerator 2 over denominator 3 left parenthesis 1 right parenthesis plus 5 end fraction close parentheses end cell row blank equals cell 0 over 0 end cell end table$

Karena jika disubstitusikan menghasilkan bentuk tak tentu maka selesaikan limit tersebut dengan metode pemfaktoran seperti berikut.

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 1 of fraction numerator 64 over denominator x minus 1 end fraction open parentheses fraction numerator 1 over denominator x plus 3 end fraction minus fraction numerator 2 over denominator 3 x plus 5 end fraction close parentheses end cell equals cell limit as x rightwards arrow 1 of fraction numerator 64 over denominator x minus 1 end fraction open parentheses fraction numerator 3 x plus 5 minus 2 left parenthesis x plus 3 right parenthesis over denominator left parenthesis x plus 3 right parenthesis left parenthesis 3 x plus 5 right parenthesis end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator 64 over denominator x minus 1 end fraction open parentheses fraction numerator x minus 1 over denominator left parenthesis x plus 3 right parenthesis left parenthesis 3 x plus 5 right parenthesis end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator 64 over denominator left parenthesis x plus 3 right parenthesis left parenthesis 3 x plus 5 right parenthesis end fraction end cell row blank equals cell fraction numerator 64 over denominator left parenthesis 1 plus 3 right parenthesis left parenthesis 3 left parenthesis 1 right parenthesis plus 5 right parenthesis end fraction end cell row blank equals cell fraction numerator 64 over denominator 4 times 8 end fraction end cell row blank equals 2 end table end style$

Jadi, nilai limit fungsi tersebut adalah 2.