Question Number 184795 by Spillover last updated on 11/Jan/23
$${Evaluate}\: \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}−\mathrm{cos}\:{x}\sqrt{\mathrm{cos}\:\mathrm{2}{x}}\:}{{x}^{\mathrm{2}} } \\ $$
Commented by MJS_new last updated on 11/Jan/23
$$\frac{\mathrm{3}}{\mathrm{2}} \\ $$
Answered by cortano1 last updated on 12/Jan/23
$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:{x}+\mathrm{cos}\:{x}−\mathrm{cos}\:{x}\sqrt{\mathrm{cos}\:\mathrm{2}{x}}}{{x}^{\mathrm{2}} } \\ $$$$=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:^{\mathrm{2}} {x}}{\left(\mathrm{1}+\mathrm{cos}\:{x}\right){x}^{\mathrm{2}} }\:+\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:{x}\left(\mathrm{1}−\sqrt{\mathrm{cos}\:\mathrm{2}{x}}\right)}{{x}^{\mathrm{2}} } \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{2}}\:+\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{cos}\:{x}\left(\mathrm{1}−\mathrm{cos}\:\mathrm{2}{x}\right)}{\left(\mathrm{1}+\sqrt{\mathrm{cos}\:\mathrm{2}{x}}\right){x}^{\mathrm{2}} } \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\:+\frac{\mathrm{1}}{\mathrm{2}}.\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2sin}\:^{\mathrm{2}} {x}}{{x}^{\mathrm{2}} } \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+\mathrm{2}\right)=\frac{\mathrm{3}}{\mathrm{2}} \\ $$