Question

Evaluate :$\underset{x\to 0}{lim}\left(\frac{1-\cos x}{x^2}\right)$

Answer 1

${lim}_{x\to 0}\frac{1-\cos x}{x^2}$

$={lim}_{x\to 0}\frac{2{\sin }^2\frac{x}{2}}{x^2}$ since $1-\cos x=2{\sin }^2\frac{x}{2}$

$={lim}_{x\to 0}\frac{2{\sin }^2\frac{x}{2}}{\frac{x^2}{4}\times 4}$ since ${lim}_{\theta \to 0}\frac{\sin \theta }{\theta }=1$

$=\frac{2}{4}\times 1=\frac{1}{2}$