Question

${lim}_{x\to 0}\frac{cos2x-1}{cosx-1}$

Answer 1

${lim}_{x\to 0}\frac{cos2x-1}{cosx-1}$

$={lim}_{x\to 0}\frac{1-cos2x}{1-cosx}={lim}_{x\to 0}\frac{2si{n}^{2}x}{2si{n}^2\frac{x}{2}}$

$=\frac{{lim}_{x\to 0}(sinx{)}^2}{{lim}_{x\to 0}\left(sin\frac{x}{2}\right)}=\frac{{lim}_{x\to 0}{\left(2sin\frac{x}{2}cos\frac{x}{2}\right)}^2}{{lim}_{x\to 0}{\left(sin\frac{x}{2}\right)}^2}$

$=4{lim}_{x\to 0}co{s}^2\frac{x}{2}=4\times 1=4$