Pertanyaan

Hitunglah nilai setiap limit berikut. b. x → 0 lim 2 + x − 2 − x 1 + x − 1 − x

Hitunglah nilai setiap limit berikut.

b. $x \to 0 lim \frac{1 + x - 1 - x}{2 + x - 2 - x}$

W. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sriwijaya

Jawaban terverifikasi

Jawaban

.

$limit as x rightwards arrow 0 of fraction numerator square root of 1 plus x end root minus square root of 1 minus x end root over denominator square root of 2 plus x end root minus square root of 2 minus x end root end fraction equals square root of 2$ .

Pembahasan

Perhatikan perhitungan berikut: Karena hasil limit tersebut ketika disubstitusikan menghasil bilangan tak tentu , maka dilakukan rasionalisasi bentuk akar sekawan sebagai berikut: Jadi, .

Perhatikan perhitungan berikut:

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of fraction numerator square root of 1 plus x end root minus square root of 1 minus x end root over denominator square root of 2 plus x end root minus square root of 2 minus x end root end fraction end cell equals cell fraction numerator square root of 1 plus 0 end root minus square root of 1 minus 0 end root over denominator square root of 2 plus 0 end root minus square root of 2 minus 0 end root end fraction end cell row blank equals cell fraction numerator square root of 1 minus square root of 1 over denominator square root of 2 minus square root of 2 end fraction end cell row blank equals cell 0 over 0 end cell end table$

Karena hasil limit tersebut ketika disubstitusikan x equals 0 menghasil bilangan tak tentu 0 over 0 , maka dilakukan rasionalisasi bentuk akar sekawan sebagai berikut:

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of fraction numerator square root of 1 plus x end root minus square root of 1 minus x end root over denominator square root of 2 plus x end root minus square root of 2 minus x end root end fraction end cell equals cell limit as x rightwards arrow 0 of fraction numerator square root of 1 plus x end root minus square root of 1 minus x end root over denominator square root of 2 plus x end root minus square root of 2 minus x end root end fraction times fraction numerator square root of 1 plus x end root plus square root of 1 minus x end root over denominator square root of 1 plus x end root plus square root of 1 minus x end root end fraction times fraction numerator square root of 2 plus x end root plus square root of 2 minus x end root over denominator square root of 2 plus x end root plus square root of 2 minus x end root end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator open parentheses up diagonal strike 1 plus x up diagonal strike negative 1 end strike plus x close parentheses open parentheses square root of 2 plus x end root plus square root of 2 minus x end root close parentheses over denominator open parentheses up diagonal strike 2 plus x up diagonal strike negative 2 end strike plus x close parentheses open parentheses square root of 1 plus x end root plus square root of 1 minus x end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator open parentheses up diagonal strike 2 x end strike close parentheses open parentheses square root of 2 plus x end root plus square root of 2 minus x end root close parentheses over denominator open parentheses up diagonal strike 2 x end strike close parentheses open parentheses square root of 1 plus x end root plus square root of 1 minus x end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator open parentheses square root of 2 plus x end root plus square root of 2 minus x end root close parentheses over denominator open parentheses square root of 1 plus x end root plus square root of 1 minus x end root close parentheses end fraction end cell row blank equals cell fraction numerator open parentheses square root of 2 plus 0 end root plus square root of 2 minus 0 end root close parentheses over denominator open parentheses square root of 1 plus 0 end root plus square root of 1 minus 0 end root close parentheses end fraction end cell row blank equals cell fraction numerator square root of 2 plus square root of 2 over denominator square root of 1 plus square root of 1 end fraction end cell row blank equals cell fraction numerator up diagonal strike 2 square root of 2 over denominator up diagonal strike 2 square root of 1 end fraction end cell row blank equals cell fraction numerator square root of 2 over denominator 1 end fraction end cell row blank equals cell square root of 2 end cell end table end style$

Jadi, $limit as x rightwards arrow 0 of fraction numerator square root of 1 plus x end root minus square root of 1 minus x end root over denominator square root of 2 plus x end root minus square root of 2 minus x end root end fraction equals square root of 2$ .