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Compute the limit of the given function

Evaluate the limit.

    \[ \lim_{x \to \pi} \frac{\log |\sin x|}{\log | \sin (2x) |}. \]


We use the trig identity \sin (2x) = 2 \sin x \cos x and then use L’Hopital’s rule to evaluate,

    \begin{align*}  \lim_{x \to \pi} \frac{\log |\sin x|}{\log |\sin (2x)|} &= \lim_{x \to \pi} \frac{\log |\sin x|}{\log |2 \sin x \cos x|} \\[9pt]  &= \lim_{x \to \pi} \frac{\log |\sin x|}{\log 2 + \log |\sin x| + \log |\cos x|} \\[9pt]  &= \lim_{x \to \pi} \frac{ \frac{\cos x}{\sin x} }{ \frac{\cos x}{\sin x} - \frac{\sin x}{\cos x}} &(\text{L'Hopital's}) \\[9pt]  &= \lim_{x \to \pi} \frac{\cos x}{\cos x - \frac{\sin x}{\cos^2 x}} \\[9pt]  &= 1.  \end{align*}

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