Nilai \( \lim _{x \rightarrow \frac{\pi}{12}} \frac{\tan 3 x}{\tan 2 x} \) adalah... A. 3 B. \( \sqrt{3} \) C. \( \frac{1}{3} \sqrt{3} \) D. \( \frac{1}{2} \sqrt{3} \) E. 1
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Untuk menentukan nilai limit \( \lim _{x \rightarrow \frac{\pi}{12}} \frac{\tan 3 x}{\tan 2 x} \), kita substitusi \( x = \frac{\pi}{12} \). 1. Hitung \( 3x \) dan \( 2x \): \[ 3x = 3 \cdot \frac{\pi}{12} = \frac{\pi}{4}, \quad 2x = 2 \cdot \frac{\pi}{12} = \frac{\pi}{6} \] 2. Sekarang kita substitusi nilai tersebut ke dalam limit: \[ \tan 3x = \tan \frac{\pi}{4} = 1 \quad \text{dan} \quad \tan 2x = \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}} \] 3. Maka, limit ini menjadi: \[ \lim _{x \rightarrow \frac{\pi}{12}} \frac{\tan 3 x}{\tan 2 x} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3} \] Dengan demikian, nilai limit tersebut adalah \( \sqrt{3} \). Jawabannya adalah B. \( \sqrt{3} \).