Question Number 192573 by senestro last updated on 21/May/23
$${solve}; \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{x}^{\mathrm{2}} {tan}\left(\frac{{sin}\pi{x}}{\mathrm{2}{x}}\right) \\ $$$${solution} \\ $$$${let}\:{L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{x}^{\mathrm{2}} {tan}\left(\frac{{sin}\pi{x}}{\mathrm{2}{x}}\right) \\ $$$${since}\:{sinx}\sim{x}−\frac{{x}^{\mathrm{3}} }{\mathrm{6}}\: \\ $$$${L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{x}^{\mathrm{2}} {tan}\left(\frac{\pi{x}}{\mathrm{2}{x}}−\frac{\pi^{\mathrm{3}} {x}^{\mathrm{3}} }{\mathrm{12}{x}}\right) \\ $$$${L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{x}^{\mathrm{2}} {tan}\left(\frac{\pi}{\mathrm{2}}−\frac{\pi^{\mathrm{3}} {x}^{\mathrm{2}} }{\mathrm{12}}\right) \\ $$$${since}\:{tan}\left(\frac{\pi}{\mathrm{2}}−{x}\right)=\frac{\mathrm{1}}{{tanx}} \\ $$$${L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{x}^{\mathrm{2}} }{{tan}\left(\frac{\pi^{\mathrm{3}} {x}^{\mathrm{2}} }{\mathrm{12}}\right)} \\ $$$${L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\frac{\pi^{\mathrm{3}} {x}^{\mathrm{2}} }{\mathrm{12}}}{{tan}\left(\frac{\pi^{\mathrm{3}} {x}^{\mathrm{2}} }{\mathrm{12}}\right)}\:\frac{\mathrm{12}}{\pi^{\mathrm{3}} } \\ $$$${L}=\frac{\mathrm{12}}{\pi^{\mathrm{3}} }\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\frac{\pi^{\mathrm{3}} {x}^{\mathrm{2}} }{\mathrm{12}}}{{tan}\left(\frac{\pi^{\mathrm{3}} {x}^{\mathrm{2}} }{\mathrm{12}}\right)} \\ $$$${since}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{x}}{{tanx}}=\mathrm{1} \\ $$$${L}=\frac{\mathrm{12}}{\pi^{\mathrm{3}} }\:\centerdot\mathrm{1}=\frac{\mathrm{12}}{\pi^{\mathrm{3}} } \\ $$$${solved}\:{by}\:{HY}\:{a}.{k}.{a}\:{senestro} \\ $$