Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

$\lim_{x \to 0}\, \left (x\cdot \cot \frac x2 \right )$ is gelijk aan	A. 1
B. 2
C.
D. 0
E.

$\lim_{x \to 0}\, \left (x\cdot \cot \frac x2 \right )$ is equal to	A. 1
B. 2
C.
D. 0
E. ∞

1^ste en 2^demanier houdt rekening met :
$\displaystyle\lim_{x\to0}\;\frac{\sin{a}x}{ax}=\lim_{x\to0}\;\frac{\tan{a}x}{ax}=\lim_{x\to0}\;\frac{ax}{\sin{a}x}=\lim_{x\to0}\;\frac{ax}{\tan{a}x}=\textbf{1}$
1^ste manier :
$\lim_{x\to0}\;\;x.\cot\frac x2=\lim_{x\to0}\;\;\frac{x.\cos{\frac{x}{2}}}{\sin{\frac{x}{2}}}=2.\lim_{x\to0}\;\frac{\frac x2}{\sin{\frac{x}{2}}}\;.\;\lim_{x\to0}\;\cos\frac x2=2.1.1=2$

2^de manier :
$\lim_{x\to0}\;\;x.\cot\frac x2=\lim_{x\to0}\;\;\frac{x}{\tan{\frac{x}{2}}}=2.\lim_{x\to0}\;\frac{\frac x2}{\tan{\frac{x}{2}}}=2.1=2$
3^de manier :
$\lim_{x\to0}\;\;x.\cot\frac x2=\lim_{x\to0}\;\;\frac{x}{\tan{\frac{x}{2}}}=\left(=\frac00\right)\buildrel H\over=\lim_{x\to0}\;\;\frac{1}{\frac{1}{\cos^2\frac x2.\frac12}}=\frac{1}{\frac11.\frac12}=2$

^{gricha - v7481 - 1.9.13}