Question Number 116136 by bemath last updated on 01/Oct/20 $$\mathrm{y}'\:=\frac{\mathrm{y}}{\mathrm{x}}\:+\frac{\mathrm{2x}^{\mathrm{3}} \:\mathrm{cos}\:\left(\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{y}} \\ $$$$\mathrm{where}\:\mathrm{y}\left(\sqrt{\pi}\right)\:=\:\mathrm{0} \\ $$ Answered by mindispower last updated on 01/Oct/20 $$\frac{{y}}{{x}}={z} \\…
Question Number 181644 by mathlove last updated on 28/Nov/22 $${prove}\:{that} \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{5}}+\centerdot\centerdot\centerdot\centerdot+\frac{\mathrm{1}}{\mathrm{2}{x}+\mathrm{1}}}{{xln}\sqrt{{x}}}\right)^{{ln}\sqrt{{x}}} =\mathrm{2}{e}^{\frac{\gamma}{\mathrm{2}}} \: \\ $$$$ \\ $$ Answered by aleks041103 last updated on…
Question Number 181602 by ali009 last updated on 27/Nov/22 $${if}\: \\ $$$$\underset{{x}\rightarrow\mathrm{1}} {{lim}}\frac{\sqrt{{ax}+{b}}−\mathrm{4}}{\left({x}−\mathrm{1}\right)}=\mathrm{5}\:\:\:\:{find}\:{a},{b} \\ $$ Answered by aleks041103 last updated on 27/Nov/22 $${as}\:{the}\:{denominator}\:{goes}\:{to}\:\mathrm{0},\:{to}\:{have} \\ $$$${a}\:{finite}\:{limit},\:{we}\:{need}\:\sqrt{{ax}+{b}}−\mathrm{4}\rightarrow\mathrm{0}…
Question Number 116043 by bemath last updated on 30/Sep/20 $$\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}^{\mathrm{4}} }}{\mathrm{x}}\:? \\ $$$$\:\:\:\int\:\mathrm{sinh}\:^{\mathrm{2}} \left(\mathrm{x}\right)\:\mathrm{cosh}\:\left(\mathrm{x}\right)\:\mathrm{dx}\: \\ $$$$\:\:\:\mathrm{find}\:\mathrm{x}\:\mathrm{from}\:\mathrm{equation}\:\mathrm{cos}\:\left(\mathrm{2tan}^{−\mathrm{1}} \left(\mathrm{x}\right)\right)=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$ Answered by MJS_new last…
Question Number 116037 by Rio Michael last updated on 30/Sep/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{27}^{{x}} −\mathrm{1}}{\mathrm{9}^{{x}} −\mathrm{1}}\:=\:?? \\ $$ Answered by bemath last updated on 30/Sep/20 $${let}\:\mathrm{3}^{{x}} =\:{t}\:;\:{t}\rightarrow\mathrm{1}…
Question Number 116039 by bobhans last updated on 30/Sep/20 $$\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\:\frac{\sqrt[{\mathrm{3}\:}]{\mathrm{cos}\:^{\mathrm{4}} \left({x}\right)}}{\left(\mathrm{1}−\mathrm{sin}\:\left({x}\right)\right)^{\frac{\mathrm{2}}{\mathrm{3}}} }\:? \\ $$ Answered by Dwaipayan Shikari last updated on 30/Sep/20 $$\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\frac{\left(\mathrm{cosx}\right)^{\frac{\mathrm{4}}{\mathrm{3}}}…
Question Number 116029 by bemath last updated on 30/Sep/20 $$\:\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:{x}}{\left(\mathrm{1}−\mathrm{sin}\:{x}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} }\:=?\: \\ $$ Answered by bobhans last updated on 30/Sep/20 $${set}\:{x}\:=\:\frac{\pi}{\mathrm{2}}\:+\:{z} \\ $$$$\underset{{z}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{−\mathrm{sin}\:{z}}{\left(\mathrm{1}−\mathrm{cos}\:{z}\right)^{\frac{\mathrm{2}}{\mathrm{3}}}…
Question Number 181553 by Frix last updated on 26/Nov/22 $$\mathrm{Reposting}\:\mathrm{question}\:\mathrm{181462} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\sqrt[{{n}}]{\sqrt{\mathrm{2}!}×\sqrt[{\mathrm{3}}]{\mathrm{3}!}×\sqrt[{\mathrm{4}}]{\mathrm{4}!}×…×\sqrt[{{n}}]{{n}!}}}{\:\sqrt[{{n}+\mathrm{1}}]{\left(\mathrm{2}{n}+\mathrm{1}\right)!!}}\overset{?} {=}\frac{\mathrm{1}}{\mathrm{2e}} \\ $$ Answered by aleks041103 last updated on 26/Nov/22 $${b}_{{n}} =\frac{\sqrt[{{n}}]{\sqrt{\mathrm{2}!}×\sqrt[{\mathrm{3}}]{\mathrm{3}!}×\sqrt[{\mathrm{4}}]{\mathrm{4}!}×…×\sqrt[{{n}}]{{n}!}}}{\:\sqrt[{{n}+\mathrm{1}}]{\left(\mathrm{2}{n}+\mathrm{1}\right)!!}}=\sqrt[{{n}}]{\frac{\sqrt{\mathrm{2}!}×\sqrt[{\mathrm{3}}]{\mathrm{3}!}×\sqrt[{\mathrm{4}}]{\mathrm{4}!}×…×\sqrt[{{n}}]{{n}!}}{\left(\left(\mathrm{2}{n}+\mathrm{1}\right)!!\right)^{\frac{{n}}{{n}+\mathrm{1}}}…
Question Number 116019 by bemath last updated on 30/Sep/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\frac{{d}}{{dx}}\:\underset{\mathrm{0}} {\overset{{x}} {\int}}\:\mathrm{sin}\:\left({t}^{\mathrm{3}} \right)\:{dt}}{\mathrm{2}{x}^{\mathrm{4}} }\:? \\ $$ Commented by bemath last updated on 30/Sep/20 $$\underset{{x}\rightarrow\mathrm{0}}…
Question Number 116005 by mnjuly1970 last updated on 30/Sep/20 $$\:\:\:\:\:\:\:\:\:…\:{advanced}\:\:{mathematics}… \\ $$$$ \\ $$$$\:\:\:\:\:{prove}\:\:{that}::: \\ $$$$\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{lim}_{{x}\rightarrow\mathrm{1}^{+} } \left(\:\zeta\left(\:{x}\:\right)\:−\frac{\mathrm{1}}{{x}\:−\:\mathrm{1}}\right)\:\overset{???} {=}\gamma\:\:\: \\ $$$$\:\:\gamma::\:\mathscr{E}{uler}\:−\:{mascheroni}\:{constant}. \\ $$$$…