Wiskunde Multiple Choice Vraag

$\displaystyle\lim_{x\to\frac{\pi}{3}}\:(\sin{x}-\sin{\frac{\pi}{3}}).tan{\frac{3x}{2}}=$	A. 0
B. \(\sqrt2\)
C. \(3\)
D. \(-\frac13\)
E. \(-\frac12\)

$\inline\displaystyle\lim_{x\to\frac{\pi}{3}}\:(\sin{x}-\sin{\frac{\pi}{3}}).tan{\frac{3x}{2}}(=0\cdot\infty)=\lim_{x\to\frac{\pi}{3}}\:\frac{\sin{x}-\sin{\frac{\pi}{3}}}{\cot\frac{3x}{2}}\;(=\frac00)\\\overset{H}{=}\lim_{x\to\frac{\pi}{3}}\:\frac{\cos x}{-\frac32.\frac{1}{\sin^2\frac{3x}{2}}}=-\frac23\cdot\lim_{x\to\frac{\pi}{3}}\:\left[\cos x.\sin^2\frac{3x}{2}\right]=-\frac23\cdot\cos\frac{\pi}{3}\cdot\sin^2\frac{\pi}{2}\\=-\frac23\cdot\frac12.1^2=-\frac13$

^{gricha - v5032 - 18.12.2025}