Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

De limiet $\displaystyle \lim_{{x \to \pm \infty}}\; \left ( 1-\frac 3x \right )^{-6x}$ is gelijk aan	A. e¹⁸
	B. e²
	C. e⁻²
	D. e⁻¹⁸
	E. e⁶

[ 6-5427 - op net sinds 22.10.14-(E)-6.11.2024 ]

Translation in E N G L I S H

The limit $https://latex.codecogs.com/png.image?\dpi{110}\displaystyle \lim_{{x \to \pm \infty}}\; \left ( 1-\frac 3x \right )^{-6x}$ is equal to	A. e¹⁸
	B. e²
	C. e⁻²
	D. e⁻¹⁸
	E. e⁶

Oplossing - Solution

$\displaystyle\lim_{{x\to\pm\infty}}\;\left(1-\frac 3x\right)^{-6x}=\displaystyle\lim_{{x\to\pm\infty}}\;\left(1+\frac3{-x}\right)^{-6x}\\=\displaystyle\lim_{{x\to\pm\infty}}\;\left(1+\frac3{-x}\right)^{\frac{-x}{3}.18}=\left[\displaystyle\lim_{{x\to\pm\infty}}\;\left(1+\frac3{-x}\right)^{\frac{-x}{3}}\right]^{18}=e^{18}$