Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

$\lim_{x \to \,+\infty}\: \left ( 1-\frac 2x \right )^{4x}=$ is gelijk aan	A. 1
	B. e^− 2
	C. e^− 8
	D. e⁸
	E.

[ 6-4369 - op net sinds 1.9.2020-(E)-6.11.2024 ]

Translation in E N G L I S H

$https://latex.codecogs.com/png.image?\dpi{110} \lim_{x \to \,\infty}\: \left ( 1-\frac 2x \right )^{4x}=$ is equal to	A. 1
	B. e^− 2
	C. e^− 8
	D. e⁸
	E. ∞

Oplossing - Solution

$\displaystyle\lim_{x\to\,+\infty}\:\left(1-\frac 2x\right)^{4x}=\displaystyle\lim_{x\to\,+\infty}\:\left(1+\frac{1}{-\frac x2}\right)^{4x}=\displaystyle\lim_{x\to\,+\infty}\:\left(1+\frac{1}{-\frac x2}\right)^{-\frac x2(-8x)}\\=\left[\displaystyle\lim_{x\to\,+\infty}\:\left(1+\frac{1}{-\frac x2}\right)^{-\frac x2}\right]^{-8x}=e^{-8x}$