Question Number 204929 by universe last updated on 02/Mar/24 $$\:\:\:\:\:\:\:\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\prod}}\:\frac{{n}^{\mathrm{2}} −{r}}{{n}^{\mathrm{2}} +{r}}\:\:=\:\:? \\ $$ Answered by witcher3 last updated on 02/Mar/24 $$\underset{\mathrm{r}=\mathrm{1}}…
Question Number 204905 by Manishkumar last updated on 02/Mar/24 $$\mathrm{4}.\:\frac{\mathrm{sin}\:\mathrm{30}^{°} \:+\:\mathrm{tan}\:\mathrm{45}^{°} \:−\:\mathrm{cosec}\:\mathrm{60}^{°} }{\mathrm{sec}\:\mathrm{30}^{°} \:+\:\mathrm{cos}\:\mathrm{60}^{°} \:+\:\mathrm{cot}\:\mathrm{45}^{°} } \\ $$$$ \\ $$$$=\:\frac{\mathrm{1}/\mathrm{2}\:+\:\mathrm{1}\:−\:\mathrm{2}/\sqrt{\mathrm{3}}}{\mathrm{2}/\sqrt{\mathrm{3}}\:+\:\mathrm{1}/\mathrm{2}\:+\:\mathrm{1}\:\:\mathrm{v}} \\ $$$$=\:\frac{\frac{\sqrt{\mathrm{3}}\:+\:\mathrm{2}\sqrt{\mathrm{3}}\:−\:\mathrm{4}}{\mathrm{2}\sqrt{\mathrm{3}}}}{\frac{\mathrm{4}\:+\:\sqrt{\mathrm{3}}\:+\:\mathrm{2}\sqrt{\mathrm{3}}}{\mathrm{2}\sqrt{\mathrm{3}}}} \\ $$$$ \\…
Question Number 204900 by universe last updated on 01/Mar/24 $$\:\:\:\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{n}!\left({e}−\mathrm{x}_{\mathrm{n}} \right)\:=\:? \\ $$$$\:\:\mathrm{where}\:\mathrm{x}_{\mathrm{n}\:} =\:\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}!}+\frac{\mathrm{1}}{\mathrm{2}!}+…+\frac{\mathrm{1}}{\mathrm{n}!} \\ $$ Commented by Frix last updated on 01/Mar/24 $${x}_{{n}}…
Question Number 204879 by universe last updated on 09/Aug/24 $$\:\:\:\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{n}!\left({e}−\mathrm{x}_{\mathrm{n}} \right)\:=\:? \\ $$$$\:\:\mathrm{where}\:\mathrm{x}_{\mathrm{n}\:} =\:\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}!}+\frac{\mathrm{1}}{\mathrm{2}!}+…+\frac{\mathrm{1}}{\mathrm{n}!} \\ $$ Commented by mr W last updated on 01/Mar/24…
Question Number 204574 by universe last updated on 22/Feb/24 $$ \\ $$$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{n}^{−\mathrm{3}/\mathrm{2}} \left[\left(\mathrm{n}+\mathrm{1}\right)^{\left(\mathrm{n}+\mathrm{1}\right)} \left(\mathrm{n}+\mathrm{2}\right)^{\left(\mathrm{n}+\mathrm{2}\right)} …\left(\mathrm{2n}\right)^{\mathrm{2n}} \right]^{\mathrm{1}/\mathrm{n}^{\mathrm{2}} } \:=\:? \\ $$$$ \\ $$ Answered by…
Question Number 204558 by universe last updated on 22/Feb/24 $$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{n}^{−\mathrm{3}/\mathrm{2}} \left[\left(\mathrm{n}+\mathrm{1}\right)^{\left(\mathrm{n}+\mathrm{1}\right)} \left(\mathrm{n}+\mathrm{2}\right)^{\left(\mathrm{n}+\mathrm{2}\right)} …\left(\mathrm{2n}\right)^{\mathrm{2n}} \right]^{\mathrm{1}/\mathrm{n}^{\mathrm{2}} } \:=\:? \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 204477 by universe last updated on 18/Feb/24 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{2n}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{x}^{\mathrm{n}} }{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx}\right)^{\mathrm{n}} =? \\ $$ Answered by witcher3 last updated on 18/Feb/24…
Question Number 204396 by Thierrybadouana last updated on 15/Feb/24 Answered by Faetmaaa last updated on 27/Feb/24 $$\mathrm{ln}\left(\mathrm{1}+{y}\right)\:\underset{{y}\rightarrow\mathrm{0}} {\sim}\:{y} \\ $$$$\mathrm{sin}\left({y}\right)\:\underset{{y}\rightarrow\mathrm{0}} {\sim}\:{y} \\ $$$$\underset{\begin{matrix}{{x}\rightarrow\mathrm{0}}\\{{x}\neq\mathrm{0}}\end{matrix}} {\mathrm{lim}}\:\frac{\mathrm{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{\mathrm{sin}^{\mathrm{2}}…
Question Number 204395 by Thierrybadouana last updated on 15/Feb/24 Answered by Lindemann last updated on 20/Feb/24 $$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }×\frac{{x}^{\mathrm{2}} }{{sin}^{\mathrm{2}} \left({x}\right)}\right)\:=\:\mathrm{1} \\ $$ Terms…
Question Number 203980 by Davidtim last updated on 03/Feb/24 $$\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\frac{{ax}^{\mathrm{2}} +{bx}+\mathrm{6}}{{x}^{\mathrm{2}} −{x}−\mathrm{2}}=\mathrm{10}\:\:;\:\:\:{find}\:\:{a}=?\:\wedge\:{b}=? \\ $$ Answered by AST last updated on 03/Feb/24 $$\frac{{a}\left({x}^{\mathrm{2}} −{x}−\mathrm{2}\right)+\left({b}+{a}\right){x}+\mathrm{6}+\mathrm{2}{a}}{{x}^{\mathrm{2}} −{x}−\mathrm{2}}=\mathrm{10}…