Akan dicari $\lim_{x\to\frac{\pi}{3}}\frac{\cos x-\cos\frac{\pi}{3}}{\tan\left(x-\frac{\pi}{3}\right)}$

Misalkan $f\left(x\right)=\cos x$ artinya $\cos\frac{\pi}{3}=f\left(\frac{\pi}{3}\right)$

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

$\lim_{x\to\frac{\pi}{3}}\frac{\cos x-\cos\frac{\pi}{3}}{\tan\left(x-\frac{\pi}{3}\right)}=\lim_{x\to\frac{\pi}{3}}\frac{f\left(x\right)-f\left(\frac{\pi}{3}\right)}{\tan\left(x-\frac{\pi}{3}\right)}$

$\Leftrightarrow\lim_{x\to\frac{\pi}{3}}\frac{\cos x-\cos\frac{\pi}{3}}{\tan\left(x-\frac{\pi}{3}\right)}=\lim_{x\to\frac{\pi}{3}}\frac{x-\frac{\pi}{3}}{\tan\left(x-\frac{\pi}{3}\right)}.\frac{f\left(x\right)-f\left(\frac{\pi}{3}\right)}{x-\frac{\pi}{3}}$

$\Leftrightarrow\lim_{x\to\frac{\pi}{3}}\frac{\cos x-\cos\frac{\pi}{3}}{\tan\left(x-\frac{\pi}{3}\right)}=\lim_{x\to\frac{\pi}{3}}\frac{x-\frac{\pi}{3}}{\tan\left(x-\frac{\pi}{3}\right)}.\lim_{x\to\frac{\pi}{3}}\frac{f\left(x\right)-f\left(\frac{\pi}{3}\right)}{x-\frac{\pi}{3}}$

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

sehingga didapat

$\lim_{x\to\frac{\pi}{3}}\frac{\cos x-\cos\frac{\pi}{3}}{\tan\left(x-\frac{\pi}{3}\right)}=1.f'\left(\frac{\pi}{3}\right)$

Karena $f\left(x\right)=\cos x$ maka $f'\left(x\right)=-\sin x$. Akibatnya

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