Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

De limiet $https://latex.codecogs.com/png.image?\large \dpi{110}\displaystyle \lim_{x \to 0}\; \frac {1-\cos x} {1-x-e^{-x}}$ is gelijk aan	A. 1
	B. − 1
	C. 0
	D.
	E.

[ 6-1822 - op net sinds 18.12.08-(E)-12.11.2023 ]

Translation in E N G L I S H

The limit $https://latex.codecogs.com/png.image?\large \dpi{110}\displaystyle \lim_{x \to 0}\; \frac {1-\cos x} {1-x-e^{-x}}$ equals	A. 1
	B. − 1
	C. 0
	D.
	E. ∞

Oplossing - Solution

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