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Compute the limit of the given function

Evaluate the limit.

    \[ \lim_{x \to \frac{\pi}{2}} \frac{\tan (3x)}{\tan x}. \]


We write \tan x = \frac{\sin x}{\cos x} and apply L’Hopital’s rule to solve

    \begin{align*}  \lim_{x \to \frac{\pi}{2}} \frac{\tan (3x)}{\tan x} &= \lim_{x \to \frac{\pi}{2}} \left( \frac{\sin (3x)}{\cos (3x)} \cdot \frac{\cos x}{\sin x} \right) \\[9pt]  &= \lim_{x \to \frac{\pi}{2}} \frac{ 3 \cos x \cos (3x) - \sin x \sin (3x)}{\cos x \cos(3x) - 3 \sin x \sin (3x)} \\[9pt]  &= \frac{1}{3}. \end{align*}

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