Question Number 197482 by cortano12 last updated on 19/Sep/23
$$\:\:\:\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{sin}^{−\mathrm{1}} \left(\frac{\mathrm{x}^{\mathrm{2}} \sqrt{\mathrm{3}}\:+\mathrm{2}}{\mathrm{2x}^{\mathrm{2}} −\mathrm{3x}+\mathrm{1}}\:\right)=?\: \\ $$
Answered by Frix last updated on 19/Sep/23
$${f}\left({x}\right)=\frac{\sqrt{\mathrm{3}}{x}^{\mathrm{2}} +\mathrm{2}}{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{1}}=\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}+\frac{\mathrm{3}\sqrt{\mathrm{3}}{x}+\mathrm{4}−\sqrt{\mathrm{3}}}{\mathrm{2}\left(\mathrm{2}{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{1}\right)} \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{f}\left({x}\right)\:=\frac{\sqrt{\mathrm{3}}}{\mathrm{2}} \\ $$$$\mathrm{sin}^{−\mathrm{1}} \:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\:=\frac{\pi}{\mathrm{3}} \\ $$