Pertanyaan

Jika x → 0 lim sin 6 x x a sin 4 x = 1 , nilai yang memenuhi ....

Jika $x \to 0 lim \frac{x ^{a} sin ^{4} x}{sin ^{6} x} = 1$ , nilai $a$ yang memenuhi ....

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A. Azriah

Master Teacher

Mahasiswa/Alumni Universitas Negeri Yogyakarta

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah B.

Pembahasan

Ingat sifat-sifat limit trigonometri berikut! x → c lim k=k Penyelesaian: Kalikan kedua ruas dengan x 6 x 6 untuk mengubah bentuk trigonometri agar sesuai dengan sifat-sifat trigonometri. maka diperoleh: x 2 x a x a − 2 x a − 2 a − 2 a = = = = = 1 1 x 0 0 2 Oleh karena itu, jawaban yang benar adalah B.

Ingat sifat-sifat limit trigonometri berikut!

$stack text lim end text with x rightwards arrow 0 below fraction numerator text sin end text x over denominator x end fraction equals 1$
$stack text lim end text with x rightwards arrow 0 below blank fraction numerator x over denominator text sin end text space x end fraction equals 1$
$x \to c lim k=k$

Penyelesaian:
Kalikan kedua ruas dengan $\frac{x ^{6}}{x ^{6}}$ untuk mengubah bentuk trigonometri agar sesuai dengan sifat-sifat trigonometri.

$table attributes columnalign right center left columnspacing 0px end attributes row cell stack text lim end text with x rightwards arrow 0 below fraction numerator x to the power of a space sin to the power of 4 space x over denominator sin to the power of 6 space x end fraction end cell equals 1 row cell stack text lim end text with x rightwards arrow 0 below open parentheses fraction numerator x to the power of a space sin to the power of 4 space x over denominator sin to the power of 6 space x end fraction times x to the power of 6 over x to the power of 6 close parentheses end cell equals cell 1. blank bold x to the power of bold 6 over bold x to the power of bold 6 end cell row cell limit as straight x rightwards arrow 0 of open parentheses fraction numerator straight x to the power of straight a space sin to the power of 4 space straight x over denominator straight x to the power of 6 end fraction times blank fraction numerator straight x to the power of 6 over denominator sin to the power of 6 space straight x end fraction close parentheses end cell equals cell 1. blank left parenthesis 1 right parenthesis end cell row cell limit as straight x rightwards arrow 0 of open parentheses fraction numerator straight x to the power of straight a space sin to the power of 4 space straight x over denominator straight x squared. straight x to the power of 4 end fraction times blank fraction numerator straight x to the power of 6 over denominator sin to the power of 6 space straight x end fraction close parentheses end cell equals 1 row cell limit as straight x rightwards arrow 0 of open parentheses straight x to the power of straight a over straight x squared times fraction numerator sin to the power of 4 space straight x over denominator straight x to the power of 4 end fraction times blank fraction numerator straight x to the power of 6 over denominator sin to the power of 6 space straight x end fraction close parentheses end cell equals 1 row cell limit as straight x rightwards arrow 0 of blank straight x to the power of straight a over straight x squared times limit as straight x rightwards arrow 0 of blank fraction numerator sin to the power of 4 space straight x over denominator straight x to the power of 4 end fraction times limit as straight x rightwards arrow 0 of blank fraction numerator straight x to the power of 6 over denominator sin to the power of 6 space straight x end fraction end cell equals 1 row cell limit as straight x rightwards arrow 0 of blank straight x to the power of straight a over straight x squared times stack text lim end text with straight x rightwards arrow 0 below blank fraction numerator text sin end text to the power of 4 text end text straight x over denominator straight x to the power of 4 end fraction times stack text lim end text with straight x rightwards arrow 0 below blank fraction numerator straight x to the power of 6 over denominator text sin end text to the power of 6 text end text straight x end fraction end cell equals 1 row cell stack text lim end text with x rightwards arrow 0 below blank x to the power of a over x squared times 1 times 1 end cell equals cell 1 space end cell row cell stack text lim end text with x rightwards arrow 0 below blank x to the power of a over x squared end cell equals 1 end table$

maka diperoleh:

$\frac{x ^{a}}{x ^{2}} x^{a - 2} x^{a - 2} a - 2 a = = = = = 11 x^{0} 02$