Question Number 221034 by Nicholas666 last updated on 23/May/25
Answered by vnm last updated on 23/May/25
$${let}\:{lim}={a} \\ $$$$\mathrm{sin}\frac{{a}}{\:\sqrt{{n}}}=\frac{{a}}{\:\sqrt{{n}}}−\frac{\mathrm{1}}{\mathrm{6}}\frac{{a}^{\mathrm{3}} }{{n}\sqrt{{n}}}+{o}\left(\frac{\mathrm{1}}{{n}\sqrt{{n}}}\right)=\frac{{a}}{\:\sqrt{{n}}}\left(\mathrm{1}−\frac{{a}^{\mathrm{2}} }{\mathrm{6}{n}}+{o}\left(\frac{\mathrm{1}}{{n}}\right)\right) \\ $$$$\sqrt{{n}+\mathrm{1}}=\sqrt{{n}}\sqrt{\mathrm{1}+\frac{\mathrm{1}}{{n}}}=\sqrt{{n}}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}{n}}+{o}\left(\frac{\mathrm{1}}{{n}}\right)\right) \\ $$$$\mathrm{sin}\frac{{a}}{\:\sqrt{{n}}}\centerdot\sqrt{{n}+\mathrm{1}}=\frac{{a}}{\:\sqrt{{n}}}\sqrt{{n}}\left(\left(\mathrm{1}−\frac{{a}^{\mathrm{2}} }{\mathrm{6}{n}}\right)\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}{n}}\right)+{o}\left(\frac{\mathrm{1}}{{n}}\right)\right)= \\ $$$${a}\left(\mathrm{1}−\frac{{a}^{\mathrm{2}} }{\mathrm{6}{n}}+\frac{\mathrm{1}}{\mathrm{2}{n}}+{o}\left(\frac{\mathrm{1}}{{n}}\right)\right) \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}−\frac{{a}^{\mathrm{2}} }{\mathrm{6}}=\mathrm{0},\:\:{a}=\sqrt{\mathrm{3}} \\ $$