Question Number 206020 by NANIGOPAL last updated on 05/Apr/24
Answered by cortano12 last updated on 05/Apr/24
$$\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}\left(\frac{\mathrm{2tan}\:{x}}{\mathrm{1}−\mathrm{tan}\:^{\mathrm{2}} {x}}\:−\mathrm{2tan}\:{x}\right)}{\mathrm{4sin}\:^{\mathrm{4}} {x}} \\ $$$$\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2}{x}\mathrm{tan}\:{x}\left(\mathrm{1}−\left(\mathrm{1}−\mathrm{tan}\:^{\mathrm{2}} {x}\right)\right)}{\mathrm{4sin}\:^{\mathrm{4}} {x}} \\ $$$$\:\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:^{\mathrm{2}} {x}}{\mathrm{sin}\:^{\mathrm{2}} {x}}\:=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$