Question Number 143960 by bobhans last updated on 20/Jun/21
$$\:\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{1}−\mathrm{cos}\:\mathrm{x}^{\mathrm{2}} }}{\mathrm{1}−\mathrm{cos}\:\mathrm{x}}\:=? \\ $$
Answered by lapache last updated on 20/Jun/21
$${li}\underset{{x}\rightarrow\mathrm{0}} {{m}}\frac{\sqrt{\mathrm{1}−\mathrm{1}+\frac{{x}^{\mathrm{4}} }{\mathrm{2}}\:}}{\mathrm{1}−\mathrm{1}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}}}={li}\underset{{x}\rightarrow\mathrm{0}} {{m}}\frac{\mathrm{2}\sqrt{\frac{{x}^{\mathrm{4}} }{\mathrm{2}}}}{{x}^{\mathrm{2}} }=\sqrt{\mathrm{2}} \\ $$
Answered by bramlexs22 last updated on 20/Jun/21
$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{2sin}\:^{\mathrm{2}} \left(\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right)}}{\mathrm{2sin}\:^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)}\:=\:\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\left(\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right)}{\mathrm{sin}\:^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)} \\ $$$$=\:\frac{\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{2}}.\:\mathrm{1}\:=\:\sqrt{\mathrm{2}}\: \\ $$