Question Number 195075 by tri26112004 last updated on 23/Jul/23 $$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{{sin}^{\mathrm{2}} {x}−{cos}^{\mathrm{3}} {x}}{{x}} \\ $$ Answered by MM42 last updated on 23/Jul/23 $$\exists\:{m}>\mathrm{0}\:;\:\forall\:{x}\in\mathbb{R}\:\Rightarrow\:\mid{sin}^{\mathrm{2}} {x}−{cos}^{\mathrm{3}} {x}\mid\leqslant{m}…
Question Number 195038 by sciencestudentW last updated on 22/Jul/23 $${prove}\:{that}\:{x}+{y}=\frac{\mathrm{1}}{{x}−{y}} \\ $$ Commented by Frix last updated on 22/Jul/23 $$\mathrm{It}'\mathrm{s}\:\mathrm{wrong},\:\mathrm{i}.\mathrm{e}.\:{x}=\mathrm{2}\wedge{y}=\mathrm{3}\:\Rightarrow\: \\ $$$${x}+{y}=\mathrm{5}\:\mathrm{but}\:\frac{\mathrm{1}}{{x}−{y}}=−\mathrm{1} \\ $$$$ \\…
Question Number 195033 by mathlove last updated on 22/Jul/23 $${any}\:{point}\:{is}\:{the}\:{function}\:{is} \\ $$$${not}\:{continous} \\ $$$${f}\left({x}\right)=\left(\mathrm{4}{x}+\mathrm{8}\right)^{\frac{{ln}\mathrm{45}}{\mathrm{8}}} \\ $$$$\left.{a}\left.\right)\:−\mathrm{8}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{b}\right)\:−\mathrm{2} \\ $$$$\left.{c}\left.\right)\:{no}\:{one}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{d}\right)\:\mathrm{5} \\ $$ Answered by alephzero last updated…
Question Number 195035 by mathlove last updated on 22/Jul/23 $$\sqrt{{i}}=? \\ $$ Answered by alephzero last updated on 22/Jul/23 $$\sqrt{{i}}\:=\:? \\ $$$$\left(\sqrt{{i}}\right)^{\mathrm{2}} \:=\:{i} \\ $$$${e}^{{i}\pi}…
Question Number 195003 by mathlove last updated on 22/Jul/23 $${y}={lnx}^{{x}^{{x}} } \:\:\:\:\:\:\:\:\:{y}^{'} =? \\ $$ Answered by shunmisaki007 last updated on 22/Jul/23 $${y}=\mathrm{ln}\left({x}^{{x}^{{x}} } \right)={x}^{{x}}…
Question Number 195002 by mathlove last updated on 22/Jul/23 $${x}^{\sqrt{{x}}} =\left(\sqrt{{x}}\right)^{{x}} \:\:\:\:\:{x}=? \\ $$ Answered by dimentri last updated on 22/Jul/23 $$\:\:\:\left({x}\right)^{\sqrt{{x}}} \:=\:\left({x}\right)^{\frac{\mathrm{1}}{\mathrm{2}}{x}} \\ $$$$\:\:\left(\mathrm{1}\right)\:\sqrt{{x}}\:=\:\frac{\mathrm{1}}{\mathrm{2}}{x}\Rightarrow{x}−\mathrm{2}\sqrt{{x}}\:=\mathrm{0}…
Question Number 195029 by cortano12 last updated on 22/Jul/23 $$\:\:\:\:\underbrace{ } \\ $$ Answered by MM42 last updated on 22/Jul/23 $${lim}_{{x}\rightarrow\mathrm{0}^{+} } \:\frac{\sqrt{\mathrm{1}+{x}}−\mathrm{1}}{\:\sqrt{\mathrm{3}}{ln}\left(\mathrm{1}+{x}\right)}\:=\overset{{hop}} {\rightarrow}\: \\…
Question Number 194998 by a.lgnaoui last updated on 22/Jul/23 $$\mathrm{Resolution}\:\mathrm{du}\:\mathrm{probldme}\:\mathrm{pose}\:\mathrm{par}\: \\ $$$$\mathrm{sonukgindia}\:\left(\mathrm{16}.\mathrm{7}.\mathrm{2023}\right) \\ $$$$\mathrm{voir}\:\:\mathrm{Q194819} \\ $$$$\bigtriangleup\boldsymbol{\mathrm{ABC}}\:\:\boldsymbol{\mathrm{AM}}=\boldsymbol{\mathrm{AN}}=\boldsymbol{\mathrm{AD}}\mathrm{cos}\:\frac{\boldsymbol{\alpha}}{\mathrm{2}} \\ $$$$\begin{cases}{\boldsymbol{\mathrm{AC}}=\boldsymbol{\mathrm{AM}}+\boldsymbol{\mathrm{MC}}=\mathrm{17}\:\:\:\:\:\:\:\:\:\:\left(\mathrm{1}\right)}\\{\boldsymbol{\mathrm{AB}}=\boldsymbol{\mathrm{AN}}+\boldsymbol{\mathrm{NB}}\:\:=\mathrm{18}\:\:\:\:\:\:\:\:\:\:\left(\mathrm{2}\right)}\end{cases} \\ $$$$\:\:\boldsymbol{\mathrm{AB}}−\boldsymbol{\mathrm{AC}}=\mathrm{1}=\boldsymbol{\mathrm{NB}}−\boldsymbol{\mathrm{MC}}\:\:\:\left(\mathrm{3}\right) \\ $$$$\bigtriangleup\boldsymbol{\mathrm{CDE}}\:\:\:\mathrm{cos}\:\frac{\boldsymbol{\mathrm{C}}}{\mathrm{2}}=\frac{\boldsymbol{\mathrm{CM}}}{\boldsymbol{\mathrm{CD}}}=\frac{\boldsymbol{\mathrm{CE}}}{\boldsymbol{\mathrm{CD}}}\:\Rightarrow\boldsymbol{\mathrm{CM}}=\boldsymbol{\mathrm{CE}} \\ $$$$\bigtriangleup\boldsymbol{\mathrm{BDE}}\:\:\:\mathrm{cos}\:\frac{\boldsymbol{\mathrm{B}}}{\mathrm{2}}=\frac{\boldsymbol{\mathrm{BE}}}{\boldsymbol{\mathrm{BD}}}=\frac{\boldsymbol{\mathrm{BN}}}{\boldsymbol{\mathrm{BD}}}\Rightarrow\boldsymbol{\mathrm{BN}}=\boldsymbol{\mathrm{BE}} \\…
Question Number 195027 by York12 last updated on 22/Jul/23 $${a}_{\mathrm{3}} {x}^{\mathrm{3}} −{x}^{\mathrm{2}} +{a}_{\mathrm{1}} {x}−\mathrm{7}=\mathrm{0}\:{is}\:{a}\:{cubic}\:{polynomial}\:{in}\:{x} \\ $$$${whose}\:{Roots}\:{are}\:\alpha\:,\:\beta\:,\:\gamma\:{positive}\:{real}\:{numbers} \\ $$$${satisfying} \\ $$$$\frac{\mathrm{225}\alpha^{\mathrm{2}} }{\alpha^{\mathrm{2}} +\mathrm{7}}=\frac{\mathrm{144}\beta^{\mathrm{2}} }{\beta^{\mathrm{2}} +\mathrm{7}}=\frac{\mathrm{100}\gamma^{\mathrm{2}} }{\gamma^{\mathrm{2}}…
Question Number 195021 by tri26112004 last updated on 22/Jul/23 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}−{sin}\:{x}}{{x}^{{n}} }=¿\:\left({n}\in{N}^{\ast} \right) \\ $$ Answered by MM42 last updated on 22/Jul/23 $${if}\:\:{x}\rightarrow\mathrm{0}\:\Rightarrow\:{x}−{sinx}\:\sim\:\frac{\mathrm{1}}{\mathrm{6}}{x}^{\mathrm{3}} \\ $$$$\Rightarrow{if}\:\:\:{n}<\mathrm{3}\:\Rightarrow\:{lim}=\mathrm{0}…