1^ste manier : zonder de regel van de l'Hospital
\(\displaystyle\lim_{{x\to 0}}\:\frac{1-\sqrt{1\,-\,\sin x}}{x}=\lim_{{x\to 0}}\:\frac{(1-\sqrt{1\,-\,\sin x}).(1+\sqrt{1\,-\,\sin x})}{x.(1+\sqrt{1\,-\,\sin x)}}\\=\lim_{{x\to 0}}\:\frac{1-1+\sin x}{x.(1+\sqrt{1\,-\,\sin x)}}\\\displaystyle=\lim_{{x\to 0}}\:\frac{\sin x}{x}.\lim_{{x\to 0}}\:\frac{1}{1+\sqrt{1\,+\,\sin x}}=1.\frac12=\frac12\)
2^de manier : met de regel van de l' Hospital
\(\displaystyle\lim_{{x\to 0}}\:\frac{1-\sqrt{1\,-\,\sin x}}{x}\left(=\frac00\right)\buildrel H\over=\lim_{{x\to 0}}\:\frac{1.(-\cos x)}{2\sqrt{1-\sin x}}=\frac12 \)