Question Number 135034 by mnjuly1970 last updated on 09/Mar/21 $$\:\:\:\:\:\:\:\:\:…{nice}\:\:\:{calculus}\:\: \\ $$$$\:\:\:\:{if}\:\:{n}\geqslant\mathrm{2}\:\:\:{and}\:\:\:{P}_{{n}} =\underset{{n}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\prod}}{sin}\left(\frac{{k}\pi}{{n}}\right) \\ $$$$\:\:\:\:\:{find}\:::\:{lim}_{{n}\rightarrow\infty} \frac{{nP}_{{n}} }{\mathrm{2}}\:\int_{\frac{\pi}{\mathrm{6}}} ^{\:\frac{\pi}{\mathrm{3}}} \frac{{cos}\left(\mathrm{3}{x}\right)}{{sin}^{{n}} \left({x}\right)}{dx} \\ $$ Terms…
Question Number 69482 by Henri Boucatchou last updated on 24/Sep/19 $$\underset{{n}\rightarrow\infty} {{lim}}\frac{\mathrm{2}+{cosn}}{\mathrm{4}{n}+{sinn}}\:=\:? \\ $$ Commented by Tony Lin last updated on 24/Sep/19 $$−\mathrm{1}\leqslant{cosn}\leqslant\mathrm{1} \\ $$$$\mathrm{1}\leqslant\mathrm{2}+{cosn}\leqslant\mathrm{3}…
Question Number 69478 by Henri Boucatchou last updated on 24/Sep/19 $$\:\:\underset{\boldsymbol{{n}}\rightarrow\infty} {\boldsymbol{{lim}sin}}\left(\boldsymbol{{n}\pi}\right)\:=\:? \\ $$ Commented by mathmax by abdo last updated on 24/Sep/19 $${sin}\left({n}\pi\right)=\mathrm{0}\:\Rightarrow{lim}_{{n}\rightarrow+\infty} {sin}\left({n}\pi\right)=\mathrm{0}…
Question Number 134937 by Study last updated on 08/Mar/21 $${li}\underset{\frac{\mathrm{1}}{{x}}\rightarrow{ln}\frac{\mathrm{1}}{\mathrm{2}}} {{m}}\frac{{ln}\mathrm{2}+{ln}\mathrm{2}\centerdot{cosx}}{{cos}^{\mathrm{2}} {ln}\sqrt{\frac{\mathrm{1}}{\mathrm{2}}}}=? \\ $$ Commented by Study last updated on 09/Mar/21 $${plz}\:{help}\:{me} \\ $$ Terms…
Question Number 69338 by Rasheed.Sindhi last updated on 22/Sep/19 Commented by Prithwish sen last updated on 22/Sep/19 $$\mathrm{it}\:\mathrm{is}\:\mathrm{the}\:\mathrm{series}\:\mathrm{of} \\ $$$$\mathrm{1}−\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{5}}−\frac{\mathrm{1}}{\mathrm{7}}+…… \\ $$$$\mathrm{tan}^{−\mathrm{1}} \mathrm{x}=\mathrm{x}−\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{3}}+\frac{\mathrm{x}^{\mathrm{5}} }{\mathrm{5}}−\frac{\mathrm{x}^{\mathrm{7}}…
Question Number 134875 by bemath last updated on 08/Mar/21 Commented by bemath last updated on 08/Mar/21 $$\mathrm{The}\:\mathrm{figure}\:\mathrm{shows}\:\mathrm{a}\:\mathrm{point}\:\mathrm{P}\:\mathrm{on}\:\mathrm{the} \\ $$$$\mathrm{parabola}\:\mathrm{y}=\mathrm{x}^{\mathrm{2}} \:\mathrm{and}\:\mathrm{the}\:\mathrm{point}\:\mathrm{Q} \\ $$$$\mathrm{where}\:\mathrm{the}\:\mathrm{perpendicular}\:\mathrm{bisector} \\ $$$$\mathrm{of}\:\mathrm{OP}\:\mathrm{intersects}\:\mathrm{the}\:\mathrm{y}−\mathrm{axis}\:.\:\mathrm{As}\: \\…
Question Number 134858 by mnjuly1970 last updated on 07/Mar/21 Answered by mathmax by abdo last updated on 07/Mar/21 $$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{lnxln}^{\mathrm{2}} \left(\mathrm{1}−\mathrm{x}\right)\:\mathrm{changement}\:\mathrm{1}−\mathrm{x}=\mathrm{t}\:\mathrm{give}\:\mathrm{x}=\mathrm{1}−\mathrm{t}\:\:\left(\mathrm{t}\rightarrow\mathrm{0}^{+} \right) \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{ln}\left(\mathrm{1}−\mathrm{t}\right)\mathrm{ln}^{\mathrm{2}} \left(\mathrm{t}\right)\:=\mathrm{g}\left(\mathrm{t}\right)\:\mathrm{we}\:\mathrm{have} \\…
Question Number 69319 by Joel122 last updated on 22/Sep/19 $$\mathrm{Show}\:\mathrm{that}\: \\ $$$$\underset{\left({x},{y}\right)\rightarrow\left(\mathrm{0},\mathrm{0}\right)} {\mathrm{lim}}\:\frac{\mathrm{3}}{{x}^{\mathrm{2}} \:+\:\mathrm{2}{y}^{\mathrm{2}} } \\ $$$$\mathrm{does}\:\mathrm{not}\:\mathrm{exist} \\ $$ Commented by Joel122 last updated on…
Question Number 69255 by TawaTawa last updated on 21/Sep/19 $$\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\:\sqrt{\frac{\mathrm{x}^{\mathrm{3}} \:−\:\mathrm{4}}{\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{3x}\:+\:\mathrm{2}}} \\ $$ Answered by Henri Boucatchou last updated on 21/Sep/19 $$\underset{\boldsymbol{{x}}\rightarrow\mathrm{2}} {\boldsymbol{{lim}}}\sqrt{\frac{\boldsymbol{{x}}^{\mathrm{3}}…
Question Number 134760 by rs4089 last updated on 07/Mar/21 Answered by mathmax by abdo last updated on 07/Mar/21 $$\mathrm{let}\:\mathrm{U}_{\mathrm{n}} =\left[\frac{\left(\mathrm{n}!\right)}{\mathrm{n}}\right]^{\frac{\mathrm{1}}{\mathrm{n}}} \:\Rightarrow\mathrm{U}_{\mathrm{n}} =\mathrm{e}^{\frac{\mathrm{1}}{\mathrm{n}}\mathrm{log}\left(\frac{\mathrm{n}!}{\mathrm{n}}\right)} \\ $$$$\mathrm{we}\:\mathrm{have}\:\mathrm{n}!\sim\mathrm{n}^{\mathrm{n}} \mathrm{e}^{−\mathrm{n}}…