Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

$\displaystyle \lim_{ x\to 0}\; \frac {x\,-\,\tan x} {x\,-\,\sin x}$ is gelijk aan	A. 1
B. 2
C. −2
D. −1
E. 0

$https://latex.codecogs.com/png.image?\dpi{110}\displaystyle \lim_{ x\to 0}\; \frac {x\,-\,\tan x} {x\,-\,\sin x}$ is equal to	A. 1
	B. 2
	C. −2
	D. −1
	E. 0

$\\\displaystyle\lim_{x\to 0}\;\frac{x\,-\,\tan x}{x\,-\,\sin x}\left(=\frac00\right)\overset{H}{=}\displaystyle\lim_{x\to 0}\;\frac{1-\frac{1}{\cos^2x}}{1-\cos x}=\displaystyle\lim_{x\to 0}\;\frac{\frac{\cos^2x-1}{\cos^2x}}{1-\cos x}=\displaystyle\lim_{x\to 0}\;\frac{(\cos x-1)(\cos x+1)}{\cos^2x\,(1-\cos x)}\\=\displaystyle\lim_{x\to 0}\;\frac{-(\cos x+1)}{\cos^2x}=\frac{-(1+1)}{1}=-2$