Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

$https://latex.codecogs.com/png.image?\dpi{110}\displaystyle \lim_{x \to 2} \: \frac{ \frac{1}{\sqrt x}- \frac{1}{\sqrt 2}}{x-2}$ is gelijk aan	A. $ \boldsymbol {1} $
	B. $ \boldsymbol {\sqrt 2} $
	C. $ \boldsymbol {\frac {\sqrt 2}{2}} $
	D. $ \boldsymbol {\frac {\sqrt 2}{8}} $
	E. $ \boldsymbol {-\:\frac {\sqrt 2}{8}} $

[ 5-3884 - op net sinds 15.2.2020-(E)-27.12.2023 ]

Translation in E N G L I S H

See above

Oplossing - Solution

$\\\lim_{x\rightarrow2}\frac{\frac{1}{\sqrt x}-\frac{1}{\sqrt2}}{x-2}=\lim_{x\rightarrow2}\frac{\frac{\sqrt2-\sqrt x}{\sqrt{2x}}}{x\!-\!2}=\lim_{x\rightarrow2}\frac{\sqrt2-\sqrt x}{\sqrt{2x}.\left(x\!-\!2\right)}=\lim_{x\rightarrow2}\frac{\sqrt2-\sqrt x}{\sqrt{2x}.\left(\sqrt x-\sqrt2\right).\left(\sqrt x+\sqrt2\right)}\\=\lim_{x\rightarrow2}\frac{-1}{\sqrt{2x}.\left(\sqrt x+\sqrt2\right)}=\frac{-1}{2.2.2.\sqrt2}=-\frac{\sqrt2}{8}\approx-0,18$

$https://latex.codecogs.com/png.image?\dpi{110}\displaystyle \lim_{x \to 2} \: \frac{ \frac{1}{\sqrt x}- \frac{1}{\sqrt 2}}{x-2}$ is gelijk aan	A. \( \boldsymbol {1} \)
	B. \( \boldsymbol {\sqrt 2} \)
	C. \( \boldsymbol {\frac {\sqrt 2}{2}} \)
	D. \( \boldsymbol {\frac {\sqrt 2}{8}} \)
	E. \( \boldsymbol {-\:\frac {\sqrt 2}{8}} \)