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I have to find its value

$\lim \limits_{n\to\infty} \frac{n}\pi \cos \left( \frac{2\pi}{3n}\right) \sin \left( \frac{4\pi}{3n}\right)$

Can you please give just clues for solving it?

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    More hints: $\dfrac{\sin({1 \over n})}{{1 \over n}}.$2012-07-30

3 Answers 3

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$1^{\rm \large st}$ clue: What happens to the cosine term as $n\to\infty$?

$2^{\rm \large nd}$ clue: Can you relate the rest of it to $\lim\limits_{x\to0}\dfrac{\sin x}{x}$ and, if so, do you know how to do that one?

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    yay thanks for the help ;)2012-07-30
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$\frac{n}{\pi}\sin \frac{4\pi}{3n}=\frac{4}{3}\,\,\frac{\sin\frac{4\pi}{3n}}{\frac{4\pi}{3n}}$

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\begin{equation*} \begin{split} \lim_{n\to\infty}\frac{n}{\pi} \cos \frac{2\pi}{3n}\sin \frac{4\pi}{3n}&= \lim_{n\to\infty} \frac{4}{3} \cos \frac{2\pi}{3n}\frac{\sin \frac{4\pi}{3n}}{\frac{4\pi}{3n}}\\ &= \frac{4}{3}.(\because \lim_{n\to\infty}\frac{\sin \frac{4\pi}{3n}}{\frac{4\pi}{3n}}=1\ \text{and} \lim_{n\to\infty}\cos \frac{2\pi}{3n}=\cos 0=1) \end{split} \end{equation*}

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    Does this count as giving "just clues"?2012-07-30