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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

Ask by Watson Lynch. in Indonesia
Mar 14,2025

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The limit is \(\sqrt{3}\), so the correct answer is B.

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Untuk mencari limit \( \lim _{x \rightarrow \frac{\pi}{12}} \frac{\tan 3 x}{\tan 2 x} \), kita akan substitusi \( x = \frac{\pi}{12} \). Pertama, kita hitung nilai \( 3x \) dan \( 2x \): \[ 3x = 3 \times \frac{\pi}{12} = \frac{\pi}{4} \] \[ 2x = 2 \times \frac{\pi}{12} = \frac{\pi}{6} \] Kemudian kita cari \( \tan 3x \) dan \( \tan 2x \): \[ \tan\left(\frac{\pi}{4}\right) = 1 \] \[ \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \] Sekarang kita substitusikan nilai tersebut ke dalam limit: \[ \lim _{x \rightarrow \frac{\pi}{12}} \frac{\tan 3 x}{\tan 2 x} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3} \] Dengan demikian, nilai limit tersebut adalah \( \sqrt{3} \). Jawaban yang tepat adalah B. \( \sqrt{3} \).

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