Pertanyaan

Nilai dari x → 0 lim 2 x 2 cos 2 x sin 2 3 x + cos 2 3 x − 1 = ...

Nilai dari $x \to 0 lim \frac{cos 2 x sin ^{2} 3 x + cos ^{2} 3 x - 1}{2 x ^{2}} = ...$

A. Allamah

Master Teacher

Mahasiswa/Alumni Universitas Brawijaya

Jawaban terverifikasi

Jawaban

x → 0 lim 2 x 2 cos 2 x sin 2 3 x + cos 2 3 x − 1 = 0 .

x \to 0 lim \frac{cos 2 x sin ^{2} 3 x + cos ^{2} 3 x - 1}{2 x ^{2}} = 0

Pembahasan

Dengan menggunakan rumus dan x → 0 lim cos a x = 1 , diperoleh = = = = = = = lim x → 0 2 x 2 c o s 2 x s i n 2 3 x + c o s 2 3 x − 1 lim x → 0 2 x 2 c o s 2 x s i n 2 3 x + lim x → 0 2 x 2 c o s 2 3 x − 1 2 1 ⋅ lim x → 0 cos 2 x ⋅ lim x → 0 x 2 s i n 2 3 x + lim x → 0 2 x 2 − s i n 2 3 x 2 1 ⋅ lim x → 0 cos 2 x ⋅ lim x → 0 x 2 s i n 2 3 x − lim x → 0 2 x 2 s i n 2 3 x 2 1 ⋅ lim x → 0 cos 2 x ⋅ lim x → 0 x s i n 3 x ⋅ lim x → 0 x s i n 3 x − lim x → 0 2 x s i n 3 x ⋅ lim x → 0 x s i n 3 x 2 1 ⋅ 1 ⋅ 3 ⋅ 3 − ( 2 3 ⋅ 3 ) 2 9 − 2 9 0 Jadi, x → 0 lim 2 x 2 cos 2 x sin 2 3 x + cos 2 3 x − 1 = 0 .

Dengan menggunakan rumus $limit as x rightwards arrow 0 of fraction numerator sin space a x over denominator b x end fraction equals a over b$ dan $x \to 0 lim cos a x = 1$ , diperoleh

$= = = = = = = lim_{x \to 0} \frac{c o s 2 x s i n ^{2} 3 x + c o s ^{2} 3 x - 1}{2 x ^{2}} lim_{x \to 0} \frac{c o s 2 x s i n ^{2} 3 x}{2 x ^{2}} + lim_{x \to 0} \frac{c o s ^{2} 3 x - 1}{2 x ^{2}} \frac{1}{2} \cdot lim_{x \to 0} cos 2 x \cdot lim_{x \to 0} \frac{s i n ^{2} 3 x}{x ^{2}} + lim_{x \to 0} \frac{- s i n ^{2} 3 x}{2 x ^{2}} \frac{1}{2} \cdot lim_{x \to 0} cos 2 x \cdot lim_{x \to 0} \frac{s i n ^{2} 3 x}{x ^{2}} - lim_{x \to 0} \frac{s i n ^{2} 3 x}{2 x ^{2}} \frac{1}{2} \cdot lim_{x \to 0} cos 2 x \cdot lim_{x \to 0} \frac{s i n 3 x}{x} \cdot lim_{x \to 0} \frac{s i n 3 x}{x} - lim_{x \to 0} \frac{s i n 3 x}{2 x} \cdot lim_{x \to 0} \frac{s i n 3 x}{x} \frac{1}{2} \cdot 1 \cdot 3 \cdot 3 - (\frac{3}{2} \cdot 3) \frac{9}{2} - \frac{9}{2} 0$