Berdasarkan rumus umum limit fungsi trigonometri bahwa

$\lim\limits_{x\to 0}\frac{\sin mx}{\tan nx}=\frac{m}{n}$

Dengan demikian,

$\lim\limits_{x\to\frac{\pi}{3}}\frac{3x\sin\left(x-\ \frac{\pi}{3}\right)}{\tan\left(\frac{1}{2}x-\frac{\pi}{6}\right)}$ $=\lim\limits_{x\to\frac{\pi}{3}}\frac{3x\sin\left(x-\frac{\pi}{3}\right)}{\tan\frac{1}{2}\left(x-\frac{\pi}{3}\right)}$

$=\lim\limits_{x\to\frac{\pi}{3}}3x\ .\ \lim\limits_{x\to\frac{\pi}{3}}\frac{\sin\left(x-\frac{\pi}{3}\right)}{\tan\frac{1}{2}\left(x-\frac{\pi}{3}\right)}$

$=\lim\limits_{x\to\frac{\pi}{3}}3x\ .\ \frac{1}{\frac{1}{2}}$

$=\lim\limits_{x\to\frac{\pi}{3}}3x\ .\ 2$

Selanjutnya, substitusikan $x=\frac{\pi}{3}$

$=3\left(\frac{\pi}{3}\right)\ .\ 2$

$=2\pi$