Pertanyaan

x → 1 lim x 3 − 3 x 2 + 2 x 7 − 2 x 5 + 1 = ...

$x \to 1 lim \frac{x ^{7} - 2 x ^{5} + 1}{x ^{3} - 3 x ^{2} + 2} = ...$

W. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sriwijaya

Jawaban terverifikasi

Jawaban

.

$limit as x rightwards arrow 1 of fraction numerator x to the power of 7 minus 2 x to the power of 5 plus 1 over denominator x cubed minus 3 x squared plus 2 end fraction equals 1$ .

Pembahasan

Perhatikan perhitungan berikut! Karena hasil dari limit tersebut ketika disubstitusikan merupakan bilangan tak tentu , maka kita lakukan pemfaktoran pada bentuk limit tersebut sebagai berikut: Jadi, .

Perhatikan perhitungan berikut!

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 1 of fraction numerator x to the power of 7 minus 2 x to the power of 5 plus 1 over denominator x cubed minus 3 x squared plus 2 end fraction end cell equals cell fraction numerator 1 to the power of 7 minus 2 times 1 to the power of 5 plus 1 over denominator 1 cubed minus 3 times 1 squared plus 2 end fraction end cell row blank equals cell fraction numerator 1 minus 2 times 1 plus 1 over denominator 1 minus 3 times 1 plus 2 end fraction end cell row blank equals cell fraction numerator 1 minus 2 plus 1 over denominator 1 minus 3 plus 2 end fraction end cell row blank equals cell fraction numerator negative 1 plus 1 over denominator negative 2 plus 2 end fraction end cell row blank equals cell 0 over 0 end cell end table$

Karena hasil dari limit tersebut ketika disubstitusikan x equals 1 merupakan bilangan tak tentu 0 over 0 , maka kita lakukan pemfaktoran pada bentuk limit tersebut sebagai berikut:

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as straight x rightwards arrow 1 of space fraction numerator straight x to the power of 7 minus 2 straight x to the power of 5 plus 1 over denominator straight x cubed minus 3 straight x squared plus 2 end fraction end cell equals cell limit as straight x rightwards arrow 1 of space fraction numerator straight x to the power of 7 minus 2 straight x to the power of 5 plus 1 over denominator straight x cubed minus 3 straight x squared plus 2 end fraction space colon space fraction numerator straight x to the power of 6 plus straight x to the power of 5 minus straight x to the power of 4 minus straight x cubed minus straight x squared minus straight x minus 1 over denominator straight x squared minus 2 straight x minus 2 end fraction end cell row cell 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 space space end cell equals cell limit as straight x rightwards arrow 1 of space fraction numerator straight x to the power of 6 plus straight x to the power of 5 minus straight x to the power of 4 minus straight x cubed minus straight x squared minus straight x minus 1 over denominator straight x squared minus 2 straight x minus 2 end fraction end cell row cell 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 space space end cell equals cell fraction numerator 1 to the power of 6 plus 1 to the power of 5 minus 1 to the power of 4 minus 1 cubed minus 1 squared minus 1 minus 1 over denominator 1 squared minus 2 open parentheses 1 close parentheses minus 2 end fraction end cell row cell 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 space space end cell equals cell fraction numerator 1 plus 1 minus 1 minus 1 minus 1 minus 1 minus 1 over denominator 1 minus 2 minus 2 end fraction end cell row cell 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 space space space end cell equals cell fraction numerator negative 3 over denominator negative 3 end fraction end cell row cell 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 space space space space end cell equals 1 end table end style$

Jadi, $limit as x rightwards arrow 1 of fraction numerator x to the power of 7 minus 2 x to the power of 5 plus 1 over denominator x cubed minus 3 x squared plus 2 end fraction equals 1$ .