Question Number 192549 by cortano12 last updated on 20/May/23
$$\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{x}^{\mathrm{2}} \:\mathrm{tan}\:\left(\frac{\mathrm{sin}\:\pi\mathrm{x}}{\mathrm{2x}}\right)\:=? \\ $$
Answered by horsebrand11 last updated on 20/May/23
$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{x}^{\mathrm{2}} \:\mathrm{tan}\:\left(\frac{\mathrm{sin}\:\pi{x}}{\mathrm{2}{x}}\right) \\ $$$$\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{x}^{\mathrm{2}} \mathrm{tan}\:\left(\frac{\pi{x}−\frac{\pi^{\mathrm{3}} {x}^{\mathrm{3}} }{\mathrm{6}}}{\mathrm{2}{x}}\right) \\ $$$$\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{x}^{\mathrm{2}} \:\mathrm{tan}\:\left(\frac{\pi}{\mathrm{2}}−\frac{\pi^{\mathrm{3}} {x}^{\mathrm{2}} }{\mathrm{12}}\right) \\ $$$$\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}^{\mathrm{2}} }{\mathrm{tan}\:\left(\frac{\pi^{\mathrm{3}} {x}^{\mathrm{2}} }{\mathrm{12}}\right)}\:=\:\frac{\mathrm{12}}{\pi^{\mathrm{3}} } \\ $$
Answered by Gamil last updated on 20/May/23
