Question Number 91897 by john santu last updated on 03/May/20
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\left(\mathrm{2}{x}−\mathrm{sin}\:{x}\right)}{\mathrm{1}−\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}\:=\:? \\ $$
Commented by john santu last updated on 03/May/20
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2}{x}−\mathrm{sin}\:{x}}{\mathrm{1}−\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right)}\:= \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2}{x}−\left({x}−\frac{{x}^{\mathrm{3}} }{\mathrm{6}}\right)}{\frac{{x}^{\mathrm{2}} }{\mathrm{2}}}\:= \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2}\left({x}+\frac{{x}^{\mathrm{3}} }{\mathrm{6}}\right)}{{x}^{\mathrm{2}} }\:= \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{2}}{{x}}+\frac{{x}}{\mathrm{3}}\:=\:{D}\mathrm{NE} \\ $$