Wiskunde Multiple Choice Vraag

$\inline\\\displaystyle\lim_{x\to\,0}\;\frac{x^4-\sin^4x}{x^6}=$	A. \(\frac16\)
B. \(\frac14\)
C. \(\frac23\)
D. \(-\frac16\)
E. \(-\frac14\)

$\inline\\\displaystyle\lim_{x\to\,0}\;\frac{x^4-\sin^4x}{x^6}=\lim_{x\to\,0}\;\frac{x^2-\sin^2x}{x^4}\cdot\lim_{x\to\,0}\;\frac{x^2+\sin^2x}{x^2}\\=(1+1^2)\cdot\lim_{x\to\,0}\;\frac{x^2-\sin^2x}{x^4}=2\cdot\lim_{x\to\,0}\;\frac{x-\sin x}{x^3}\cdot\lim_{x\to\,0}\;\frac{x+\sin x}{x}\\=2\cdot(1+1)\cdot\lim_{x\to\,0}\;\frac{x+\sin x}{x^3}=4\cdot\lim_{x\to\,0}\;\frac{x+\sin x}{x^3}\\\left(=\frac00\right)\overset{H}{=}4\cdot\lim_{x\to\,0}\;\frac{1-\cos x}{3x^2}\overset{H}{=}4\lim_{x\to\,0}\;\frac{\sin x}{6x}=4\cdot\frac16=\frac46=\frac23$

^{gricha - vA166 - 31.3.2026}