Akan dicari $\lim_{x\to\frac{\pi}{6}}\frac{\csc x-\csc\frac{\pi}{6}}{\sin\left(x-\frac{\pi}{6}\right)}$

Misalkan $f\left(x\right)=\csc x$ artinya $\csc\frac{\pi}{6}=f\left(\frac{\pi}{6}\right)$

Perlu diingat bahwa untuk sembarang fungsi $f\left(x\right)$ dan $g\left(x\right)$ berlaku

$\lim_{x\to c}f\left(x\right)g\left(x\right)=\lim_{x\to c}f\left(x\right)\lim_{x\to c}g\left(x\right)$

Berdasarkan yang diketahui pada soal dan pemisalan yang dibuat, diperoleh

$\lim_{x\to\frac{\pi}{6}}\frac{\csc x-\csc\frac{\pi}{6}}{\sin\left(x-\frac{\pi}{6}\right)}=\lim_{x\to\frac{\pi}{6}}\frac{f(x)-f(\frac{\pi}{6})}{\sin\left(x-\frac{\pi}{6}\right)}$

$\Leftrightarrow\lim_{x\to\frac{\pi}{6}}\frac{\csc x-\csc\frac{\pi}{6}}{\sin\left(x-\frac{\pi}{6}\right)}=\lim_{x\to\frac{\pi}{6}}\frac{x-\frac{\pi}{6}}{\sin\left(x-\frac{\pi}{6}\right)}\cdot\frac{f(x)-f(\frac{\pi}{6})}{x-\frac{\pi}{6}}$

$\Leftrightarrow\lim_{x\to\frac{\pi}{6}}\frac{\csc x-\csc\frac{\pi}{6}}{\sin\left(x-\frac{\pi}{6}\right)}=\lim_{x\to\frac{\pi}{6}}\frac{x-\frac{\pi}{6}}{\sin\left(x-\frac{\pi}{6}\right)}\cdot\lim_{x\to\frac{\pi}{6}}\frac{f(x)-f(\frac{\pi}{6})}{x-\frac{\pi}{6}}$

Perlu diingat pula bahwa $\lim_{x\to c}\frac{x-a}{\sin\left(x-a\right)}=1$ dan $\lim_{x\to c}\frac{f\left(x\right)-f\left(c\right)}{x-c}=f'\left(c\right)$

sehingga didapat

$\lim_{x\to\frac{\pi}{6}}\frac{\csc x-\csc\frac{\pi}{6}}{\sin\left(x-\frac{\pi}{6}\right)}=1.f'\left(\frac{\pi}{6}\right)$

Karena $f\left(x\right)=\csc x$ maka $f'\left(x\right)=-\csc x\cot x$. Akibatnya

$f'(\frac{\pi}{6})=-\csc(\frac{\pi}{6})\cot(\frac{\pi}{6})$

$f'(\frac{\pi}{6})=-\frac{1}{\sin(\frac{\pi}{6})}\frac{\cos\left(\frac{\pi}{6}\right)}{\sin\left(\frac{\pi}{6}\right)}$

$f'(\frac{\pi}{6})=-\frac{1}{\frac{1}{2}}\frac{\frac{1}{2}\sqrt{3}}{\frac{1}{2}}$

$f'(\frac{\pi}{6})=-\frac{\sqrt{3}}{\frac{1}{2}}$

$f'(\frac{\pi}{6})=-2\sqrt{3}$