Mathematics - Limits Question with Solution | TestHub

$1-\cos (1-\cos x)=2{\sin }^2\left(\frac{1-\cos x}{2}\right)$ $=2{\sin }^2\left({\sin }^2\frac{x}{2}\right)$ $\therefore {\lim }_{x\to 0}\frac{1-\cos (1-\cos x)}{x^4}={\lim }_{x\to 0}\frac{2{\sin }^2\left({\sin }^2\frac{x}{2}\right)}{x^4}$ $={\lim }_{x\to 0}\frac{2{\sin }^2\left({\sin }^2\frac{x}{2}\right)}{{\left({\sin }^2\frac{x}{2}\right)}^2}\times \frac{{\sin }^4\frac{x}{2}}{{\left(\frac{x}{2}\right)}^4\times 16}$ $=\frac{1}{8}$