Wiskunde Multiple Choice Vraag

$\displaystyle\lim_{x\to\,0}\:\frac{x^3-\sin^3x}{x^5}=$	A. −1
B. 1
C. \(\boldsymbol{\frac12}\)
D. \(\boldsymbol{\frac35}\)
E. \(\boldsymbol{\frac15}\)
F. \(\boldsymbol{\frac16}\)

We houden rekening met a³ − b³ = (a − b)(a² + ab + b²)
$\inline\\\displaystyle\lim_{x\to\,0}\:\frac{x^3-\sin^3x}{x^5}=\lim_{x\to\,0}\:\frac{x-\sin x}{x^3}\cdot\lim_{x\to\,0}\:\frac{x^3+x\sin x+\sin^2x}{x^2}\\\overset{H}{=}\lim_{x\to\,0}\:\frac{1-\cos x}{3x^2}\cdot\left(\lim_{x\to\,0}\:x\;+\lim_{x\to\,0}\:\frac{\sin x}{x}+\left(\lim_{x\to\,0}\:\frac{\sin x}{x}\right)^2\right)\\\overset{H}{=}\lim_{x\to\,0}\:\frac{\sin x}{6x}\cdot(1+1+1^2)=\frac36=\frac12$

^{gricha - vA152 - 30.3.2026}