Tentukan nilai imit masing-masing fungsi rasional berikut. e. x→4lim4x2−25x+363x2−17x+20

Pertanyaan

Tentukan nilai imit masing-masing fungsi rasional berikut.

e. $begin mathsize 14px style limit as x rightwards arrow 4 of fraction numerator 3 x squared minus 17 x plus 20 over denominator 4 x squared minus 25 x plus 36 end fraction end style$

W. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sriwijaya

Jawaban terverifikasi

Jawaban

$limit as x rightwards arrow 4 of fraction numerator 3 x squared minus 17 x plus 20 over denominator 4 x squared minus 25 x plus 36 end fraction equals 1$ .

Pembahasan

Perhatikan perhitungan berikut:

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 4 of fraction numerator 3 x squared minus 17 x plus 20 over denominator 4 x squared minus 25 x plus 36 end fraction end cell equals cell fraction numerator 3 times 4 squared minus 17 times 4 plus 20 over denominator 4 times 4 squared minus 25 times 4 plus 36 end fraction end cell row blank equals cell fraction numerator 3 times 16 minus 17 times 4 plus 20 over denominator 4 times 16 minus 25 times 4 plus 36 end fraction end cell row blank equals cell fraction numerator 48 minus 68 plus 20 over denominator 64 minus 100 plus 36 end fraction end cell row blank equals cell 0 over 0 end cell end table$

Karena hasil limit ketika disubstitusikan nilai x equals 4 menghasilkan bilangan tak tentu 0 over 0 , maka nilai limit ditentukan dengan cara pemfaktoran sebagi berikut:

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 4 of fraction numerator 3 x squared minus 17 x plus 20 over denominator 4 x squared minus 25 x plus 36 end fraction end cell equals cell limit as x rightwards arrow 4 of fraction numerator open parentheses 3 x minus 5 close parentheses up diagonal strike open parentheses x minus 4 close parentheses end strike over denominator open parentheses 4 x minus 9 close parentheses up diagonal strike open parentheses x minus 4 close parentheses end strike end fraction end cell row blank equals cell limit as x rightwards arrow 4 of fraction numerator open parentheses 3 x minus 5 close parentheses over denominator open parentheses 4 x minus 9 close parentheses end fraction end cell row blank equals cell fraction numerator 3 times 4 minus 5 over denominator 4 times 4 minus 9 end fraction end cell row blank equals cell fraction numerator 12 minus 5 over denominator 16 minus 9 end fraction end cell row blank equals cell 7 over 7 end cell row blank equals 1 end table$

Jadi, $limit as x rightwards arrow 4 of fraction numerator 3 x squared minus 17 x plus 20 over denominator 4 x squared minus 25 x plus 36 end fraction equals 1$ .