Question Number 142721 by mathlove last updated on 04/Jun/21
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{x}}{\:\sqrt{\mathrm{1}−\mathrm{cos}\:{x}}}=? \\ $$
Answered by Ar Brandon last updated on 04/Jun/21
$$\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{x}}{\:\sqrt{\mathrm{1}−\mathrm{cosx}}}=\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{x}}{\:\sqrt{\mathrm{1}−\left(\mathrm{1}−\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}\right)}}=\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{x}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} /\mathrm{2}}}=\frac{\mathrm{x}\sqrt{\mathrm{2}}}{\pm\mathrm{x}}=\pm\sqrt{\mathrm{2}} \\ $$