回复 3# isee
\begin{align*}
&\,\lim\limits_{n\to+\infty}3n\left( \sin\frac{\pi}{6n}+\tan \frac{\pi}{6n} \right)\\
=&\lim\limits_{n\to+\infty}3n\sin\frac{\pi}{6n}\left(1+\dfrac{\tan \frac{\pi}{6n}}{ \sin\frac{\pi}{6n}} \right)\\
=&\lim\limits_{n\to+\infty}\dfrac{\sin\frac{\pi}{6n}}{\frac{1}{3n}}\left(1+\dfrac{\tan \frac{\pi}{6n}}{ \sin\frac{\pi}{6n}} \right)\\
=&\lim\limits_{t\to0}\dfrac{\sin\left(\frac{\pi}{6}t\right)}{\frac{1}{3}t}\left[1+\dfrac{\tan\left(\frac{\pi}{6}t\right)}{ \sin\left(\frac{\pi}{6}t\right)} \right]\\
=&\lim\limits_{t\to0}\dfrac{\sin\left(\frac{\pi}{6}t\right)}{\frac{2}{\pi}\frac{\pi}{6}t}\left[1+\dfrac{\tan\left(\frac{\pi}{6}t\right)}{ \sin\left(\frac{\pi}{6}t\right)} \right]=\dfrac{1}{\frac{2}{\pi}}\cdot2=\pi\\
\end{align*} |