Pertanyaan

x → 0 lim x 1 + sin x − 1 − sin x = ...

$x \to 0 lim \frac{1 + sin x - 1 - sin x}{x} = ...$

N. Puspita

Master Teacher

Jawaban terverifikasi

Pembahasan

Ingat kembali faktorisasi aljabar, sifat limitdan limit fungsi trigonometri berikut. Dari aturan di atas, maka diperoleh = = = = = = = = = = lim x → 0 x 1 + s i n x − 1 − s i n x lim x → 0 ( x 1 + s i n x − 1 − s i n x ) ( 1 + s i n x + 1 − s i n x 1 + s i n x + 1 − s i n x ) lim x → 0 x ( 1 + s i n x + 1 − s i n x ) ( 1 + s i n x ) 2 − ( 1 − s i n x ) 2 lim x → 0 x ( 1 + s i n x + 1 − s i n x ) ( 1 + s i n x ) − ( 1 − s i n x ) lim x → 0 x ( 1 + s i n x + 1 − s i n x ) 2 s i n x lim x → 0 x s i n x ⋅ lim x → 0 1 + s i n x + 1 − s i n x 2 1 ⋅ 1 + s i n 0 + 1 − s i n 0 2 1 ⋅ 1 + 0 + 1 − 0 2 1 ⋅ 1 + 1 2 1 ⋅ 2 2 1 Dengan demikian, . Jadi, jawaban yang benar adalah A.

Ingat kembali faktorisasi aljabar, sifat limit dan limit fungsi trigonometri berikut.

$limit as x rightwards arrow 0 of fraction numerator sin space a x over denominator b x end fraction equals limit as x rightwards arrow 0 of fraction numerator space a x over denominator sin space b x end fraction equals a over b$

Dari aturan di atas, maka diperoleh

$= = = = = = = = = = lim_{x \to 0} \frac{1 + s i n x - 1 - s i n x}{x} lim_{x \to 0} (\frac{1 + s i n x - 1 - s i n x}{x}) (\frac{1 + s i n x + 1 - s i n x}{1 + s i n x + 1 - s i n x}) lim_{x \to 0} \frac{( 1 + s i n x ) ^{2} - ( 1 - s i n x ) ^{2}}{x ( 1 + s i n x + 1 - s i n x )} lim_{x \to 0} \frac{( 1 + s i n x ) - ( 1 - s i n x )}{x ( 1 + s i n x + 1 - s i n x )} lim_{x \to 0} \frac{2 s i n x}{x ( 1 + s i n x + 1 - s i n x )} lim_{x \to 0} \frac{s i n x}{x} \cdot lim_{x \to 0} \frac{2}{1 + s i n x + 1 - s i n x} 1 \cdot \frac{2}{1 + s i n 0 + 1 - s i n 0} 1 \cdot \frac{2}{1 + 0 + 1 - 0} 1 \cdot \frac{2}{1 + 1} 1 \cdot \frac{2}{2} 1$

Dengan demikian, $limit as x rightwards arrow 0 of fraction numerator square root of 1 plus sin space x end root minus square root of 1 minus sin space x end root over denominator x end fraction equals 1$ .

Jadi, jawaban yang benar adalah A.