Question Number 116331 by bobhans last updated on 03/Oct/20 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{3x}\left(\mathrm{cos}\:\mathrm{7x}−\mathrm{cos}\:\mathrm{3x}\right)}{\:\sqrt{\mathrm{tan}\:\mathrm{2x}+\mathrm{1}}−\sqrt{\mathrm{sin}\:\mathrm{2x}+\mathrm{1}}}\:? \\ $$ Answered by bemath last updated on 03/Oct/20 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\sqrt{\mathrm{tan}\:\mathrm{2x}+\mathrm{1}}+\sqrt{\mathrm{sin}\:\mathrm{2x}+\mathrm{1}}\right)×\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{3x}\left(−\mathrm{2sin}\:\mathrm{5x}.\mathrm{sin}\:\mathrm{2x}\right)}{\mathrm{tan}\:\mathrm{2x}\left(\mathrm{1}−\mathrm{cos}\:\mathrm{2x}\right)} \\ $$$$=\:\mathrm{2}\:×\underset{{x}\rightarrow\mathrm{0}}…
Question Number 116317 by bemath last updated on 03/Oct/20 $$\:\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{\sqrt{\mathrm{n}+\mathrm{1}}+\sqrt{\mathrm{n}+\mathrm{2}}+\sqrt{\mathrm{n}+\mathrm{3}}+…+\sqrt{\mathrm{2n}−\mathrm{1}}}{\mathrm{n}^{\frac{\mathrm{3}}{\mathrm{2}}} }\:\right)\:= \\ $$ Answered by Bird last updated on 03/Oct/20 $${U}_{{n}} \:=\frac{\mathrm{1}}{{n}^{\frac{\mathrm{3}}{\mathrm{2}}} }\sum_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}}…
Question Number 116252 by bemath last updated on 02/Oct/20 $$\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\:\frac{\mathrm{e}^{\mathrm{x}} +\mathrm{e}^{−\mathrm{x}} }{\mathrm{e}^{\mathrm{x}} −\mathrm{e}^{−\mathrm{x}} }\:=\:? \\ $$ Answered by MJS_new last updated on 02/Oct/20 $$\frac{\mathrm{e}^{{x}}…
Question Number 116167 by Ar Brandon last updated on 01/Oct/20 $$\mathrm{Show}\:\mathrm{that}\:\forall{n}\geqslant\mathrm{2}\:\mathrm{the}\:\mathrm{equation}\:{x}^{{n}} ={x}+{n} \\ $$$$\mathrm{admits}\:\mathrm{a}\:\mathrm{unique}\:\mathrm{solution}\:\mathrm{u}_{\mathrm{n}} \in\left(\mathrm{1},\mathrm{2}\right] \\ $$ Answered by 1549442205PVT last updated on 01/Oct/20 $$\mathrm{Put}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{n}}…
Question Number 116166 by Ar Brandon last updated on 01/Oct/20 $$\forall{n}\in\mathbb{N}^{\ast} ,\:\mathrm{suppose}\:{u}_{{n}} =\left(\mathrm{5sin}\frac{\mathrm{1}}{{n}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{5}}\mathrm{cos}\:{n}\right)^{{n}} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\underset{{n}\rightarrow+\infty} {\mathrm{lim}}{u}_{{n}} =\mathrm{0} \\ $$ Answered by mindispower last updated…
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}…