Question Number 205580 by universe last updated on 25/Mar/24 $$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\left(\mathrm{2}{n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{3}\right)…\left(\mathrm{4}{n}+\mathrm{1}\right)}{\left(\mathrm{2}{n}\right)\left(\mathrm{2}{n}+\mathrm{2}\right)…\left(\mathrm{4}{n}\right)}\:\:=\:\:? \\ $$ Commented by lepuissantcedricjunior last updated on 26/Mar/24 $$\underset{\boldsymbol{{n}}\rightarrow\infty} {\mathrm{lim}}\frac{\left(\mathrm{2}\boldsymbol{{n}}+\mathrm{1}\right)×\left(\mathrm{2}\boldsymbol{{n}}+\mathrm{3}\right)×…×\left(\mathrm{4}\boldsymbol{{n}}+\mathrm{1}\right)}{\left(\mathrm{2}\boldsymbol{{n}}\right)×\left(\mathrm{2}\boldsymbol{{n}}+\mathrm{2}\right)×..×\left(\mathrm{4}\boldsymbol{{n}}\right)} \\ $$$$=\underset{\boldsymbol{{n}}\rightarrow\infty} {\mathrm{lim}}\frac{\left(\mathrm{2}\boldsymbol{{n}}\right)\left(\mathrm{2}\boldsymbol{{n}}+\mathrm{1}\right)\left(\mathrm{2}\boldsymbol{{n}}+\mathrm{2}\right)…\left(\mathrm{4}\boldsymbol{{n}}\right)\left(\mathrm{4}\boldsymbol{{n}}+\mathrm{1}\right)}{\left[\left(\mathrm{2}\boldsymbol{{n}}\right)\left(\mathrm{2}\boldsymbol{{n}}+\mathrm{2}\right)….\left(\mathrm{4}\boldsymbol{{n}}\right)\right]^{\mathrm{2}}…
Question Number 205448 by mathlove last updated on 21/Mar/24 $${A}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{sinx}}{{x}^{\mathrm{3}} }=? \\ $$ Answered by namphamduc last updated on 21/Mar/24 $${A}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\left({x}\right)}{{x}^{\mathrm{3}} }=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\left({x}\right)}{{x}}.{x}^{\mathrm{4}}…
Question Number 205379 by cortano12 last updated on 19/Mar/24 Answered by MM42 last updated on 19/Mar/24 $${hop}\rightarrow={lim}_{{x}\rightarrow\mathrm{0}} \frac{−{cosx}×{sin}\left({sinx}\right)+{sinx}}{\mathrm{4}{x}^{\mathrm{3}} } \\ $$$${hop}\rightarrow=\frac{{sinx}×{sin}\left({sinx}\right)−{cos}^{\mathrm{2}} {x}×{cos}\left({sinx}\right)+{cosx}}{\mathrm{12}{x}^{\mathrm{2}} } \\ $$$$={lim}_{{x}\rightarrow\mathrm{0}}…
Question Number 205339 by depressiveshrek last updated on 17/Mar/24 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\mathrm{tan}\left(\mathrm{tan}{x}\right)}{\mathrm{sin}\left(\mathrm{1}−\mathrm{cos}{x}\right)} \\ $$ Answered by MM42 last updated on 18/Mar/24 $$={lim}_{{x}\rightarrow\mathrm{0}} \:\frac{{tan}\left({x}\right)}{{sin}\left(\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} \right)} \\ $$$$={lim}_{{x}\rightarrow\mathrm{0}}…
Question Number 205307 by universe last updated on 15/Mar/24 $$\:\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\lfloor{a}\rfloor+\lfloor\mathrm{2}{a}\rfloor+…+\lfloor{na}\rfloor}{{n}^{\mathrm{2}} }\:\mathrm{where}\:{a}\in\mathbb{R} \\ $$$$\:\:\:\mathrm{and}\:\lfloor{x}\rfloor\:\mathrm{is}\:\mathrm{the}\:\mathrm{floor}\:\mathrm{of}\:\mathrm{x}\:\in\:\mathbb{R} \\ $$ Commented by Frix last updated on 15/Mar/24 $$\mathrm{Just}\:\mathrm{guessing}: \\…
Question Number 205237 by universe last updated on 13/Mar/24 Answered by Berbere last updated on 13/Mar/24 $${n}^{\mathrm{2}} +{x}^{\mathrm{2}} \geqslant{n}^{\mathrm{2}} \\ $$$$\frac{{x}}{\mathrm{1}+{x}}\leqslant\mathrm{1}\Rightarrow\frac{{nx}\mathrm{tan}^{−\mathrm{1}} \left({x}\right)}{\left(\mathrm{1}+{x}\right)\left({n}^{\mathrm{2}} +{x}^{\mathrm{2}} \right)}\leqslant{n}.\mathrm{1}.\frac{\mathrm{tan}^{−\mathrm{1}} \left({x}\right)}{{n}^{\mathrm{2}}…
Question Number 205142 by universe last updated on 10/Mar/24 $$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{n}^{−\mathrm{n}^{\mathrm{2}} } \left[\left(\mathrm{n}+\mathrm{1}\right)\left(\mathrm{n}+\frac{\mathrm{1}}{\mathrm{2}}\right)\left(\mathrm{n}+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }\right)…\left(\mathrm{n}+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{n}−\mathrm{1}} }\right)\right]^{\mathrm{n}} =? \\ $$ Answered by pi314 last updated on 10/Mar/24…
Question Number 205134 by universe last updated on 09/Mar/24 Answered by pi314 last updated on 09/Mar/24 $${nx}={y} \\ $$$$\Leftrightarrow{A}=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\int_{\mathrm{0}} ^{{n}} \frac{{f}\left(\frac{{y}}{{n}}\right)}{\left(\mathrm{1}+{y}^{\mathrm{2}} \right)}{dy}=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\int_{\mathrm{0}} ^{{n}}…
Question Number 205114 by 2kdw last updated on 09/Mar/24 $${Solve}: \\ $$$$ \\ $$$$\:\:{lim}_{\left({x},{y}\right)\rightarrow\left(\mathrm{0},\mathrm{0}\right)} \frac{\mathrm{1}−{cos}\left(\sqrt{\mathrm{10}{xy}}\right)}{\mathrm{3}.{y}.{sin}\left(\mathrm{22}{x}\right)} \\ $$$$ \\ $$$${Ans}.:\:\frac{\mathrm{5}}{\mathrm{66}} \\ $$$${Step}\:{by}\:{step},\:{please}! \\ $$ Answered by…
Question Number 204991 by tigrecomplexe last updated on 04/Mar/24 Commented by tigrecomplexe last updated on 05/Mar/24 $${sommeone}\:{can}\:{put}\:{hand}\:{here}\:{please}\:? \\ $$ Answered by witcher3 last updated on…