Question

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

Answer 1

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

We know that $\cos x=1-2{\sin }^2\frac{x}{2}$

$\Rightarrow {lim}_{x\to 0}\left(\frac{1-(1-2{\sin }^2\frac{x}{2})}{x^2}\right)$

$\Rightarrow {lim}_{\frac{x}{2}\to 0}\frac{2{\sin }^2\frac{x}{2}}{{\left(\frac{x}{2}\right)}^2\times 4}$

We know that ${lim}_{\theta \to 0}\frac{\sin \theta }{\theta }=1$

$\therefore {lim}_{x\to 0}\left(\frac{1-\cos x}{x^2}\right)=2\times 1\times \frac{1}{4}=\frac{1}{2}$ since ${lim}_{\frac{x}{2}\to 0}\frac{\sin \frac{x}{2}}{\frac{x}{2}}=1$