Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

$\displaystyle \lim_{x \to +\infty}\left ( \frac{x\,-\,3}{x} \right )^{\frac x2}$ is gelijk aan	A. 1
B. e^1,5
C. e⁻³
D. e^−1,5
E.

$\displaystyle \lim_{x \to \infty}\left ( \frac{x\,-\,3}{x} \right )^{\frac x2}$ equals	A. 1
	B. e^1.5
	C. e⁻³
	D. e^−1.5
	E. ∞

$\displaystyle\lim_{x\to+\infty}\left(\frac{x\,-\,3}{x}\right)^{\frac x2}=\displaystyle\lim_{x\to+\infty}\left(1+\frac{3}{-x}\right)^{\frac x2}=\displaystyle\lim_{x\to+\infty}\left(1+\frac{3}{-x}\right)^{\frac{-x}{3}.\frac{-x}{2}}$
$\\=\!\!\displaystyle\lim_{x\to+\infty}\left(\left(1+\frac{3}{-x}\right)^{\frac{-x}{3}}\right)^{-\frac32}\!\!=\left(\displaystyle\lim_{x\to+\infty}\left(1+\frac{3}{-x}\right)^{\frac{-x}{3}}\right)^{-\frac32}\!\!=e^{-\frac32}\!\!=e^{-1,5}=\frac{1}{e\sqrt e$