s'il vous plait je veux savoir la démonstration de lim1-cos(x)/x ² lorsque x tend vers 0

Bonjour Lema368

[tex]\lim_{x\to0}\dfrac{1-\cos(x)}{x^2}\\\\=\lim_{x\to0}\dfrac{2\sin^2(\dfrac{x}{2})}{x^2}\\\\=\lim_{x\to0}\dfrac{\sin^2(\dfrac{x}{2})}{\dfrac{x^2}{2}}\\\\=\lim_{x\to0}\dfrac{\sin^2(\dfrac{x}{2})}{2\times\dfrac{x^2}{4}}\\\\=\lim_{x\to0}\dfrac{\sin^2(\dfrac{x}{2})}{2\times(\dfrac{x}{2})^2}[/tex]

[tex]=\lim_{x\to0}\dfrac{1}{2}\times\dfrac{\sin(\dfrac{x}{2})}{\dfrac{x}{2}}\times\dfrac{\sin(\dfrac{x}{2})}{\dfrac{x}{2}}\\\\=\lim_{x\to0}\dfrac{1}{2}\times\lim_{x\to0}\dfrac{\sin(\dfrac{x}{2})}{\dfrac{x}{2}}\times\lim_{x\to0}\dfrac{\sin(\dfrac{x}{2})}{\dfrac{x}{2}}\\\\=\dfrac{1}{2}\times1\times1\\\\=\dfrac{1}{2}[/tex]

Par conséquent,

[tex]\boxed{\lim_{x\to0}\dfrac{1-\cos(x)}{x^2}=\dfrac{1}{2}}[/tex]