Question Number 86031 by arcana last updated on 26/Mar/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{2}}−\sqrt{\mathrm{1}+\mathrm{cos}\:{x}}}{\mathrm{sin}\:^{\mathrm{2}} {x}}= \\ $$ Commented by arcana last updated on 26/Mar/20 $${the}\:{answer}\:{is}\:\frac{\mathrm{1}}{\mathrm{4}\sqrt{\mathrm{2}}} \\ $$ Commented…
Question Number 151559 by peter frank last updated on 21/Aug/21 $$\mathrm{prove}\:\mathrm{that} \\ $$$$\left(\mathrm{1}+\mathrm{cos}\:\theta+\mathrm{isin}\:\theta\right)^{\mathrm{n}} \\ $$$$+\:\left(\mathrm{1}+\mathrm{cos}\:\theta−\mathrm{isin}\:\theta\right)^{\mathrm{n}} =\mathrm{2}^{\mathrm{n}+\mathrm{1}} \mathrm{cos}\:\frac{\theta}{\mathrm{2}}\mathrm{cos}\:\frac{\mathrm{n}\theta}{\mathrm{2}} \\ $$ Answered by Olaf_Thorendsen last updated on…
Question Number 151538 by DELETED last updated on 21/Aug/21 Answered by DELETED last updated on 21/Aug/21 $$\left.\mathrm{3}\right).\:\underset{\mathrm{t}\rightarrow\infty} {\mathrm{lim}}\:\left[\left(\mathrm{sin}\:\frac{\mathrm{2}}{\mathrm{t}}\right)−\frac{\mathrm{3}}{\mathrm{t}}\right].\frac{\mathrm{t}}{\mathrm{6}}=..? \\ $$$$\:\:\:\:\:\:=\underset{\mathrm{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left[\left(\mathrm{sin}\:\mathrm{2t}\right)−\mathrm{3t}\right].\frac{\mathrm{1}}{\mathrm{6t}} \\ $$$$\:\:\:\:\:\:=\underset{\mathrm{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left[\left(\frac{\mathrm{sin}\:\mathrm{2t}}{\mathrm{6t}}\right)−\frac{\mathrm{1}}{\mathrm{2}}\right] \\…
Question Number 85985 by Rio Michael last updated on 26/Mar/20 $$\mathrm{Consider}\:\mathrm{the}\:\mathrm{function}{f}\:\mathrm{defined}\:\mathrm{by}\:\mathrm{par}{f}\left({x}\right)\:=\:−{x}\:+\:\frac{\mathrm{ln}\:{x}}{{x}}\:\mathrm{in}\:\mathrm{the}\:\mathrm{interval} \\ $$$$\left.:\:\right]\mathrm{0},+\infty\left[.\:\:\left({C}_{{f}} \right)\:\mathrm{is}\:\mathrm{its}\:\mathrm{representative}\:\mathrm{curve}\:\mathrm{in}\:\mathrm{an}\:\mathrm{orthonormal}\right. \\ $$$$\mathrm{reference}\:\mathrm{system}\:\left(\mathrm{O},\overset{\rightarrow} {{i}},\overset{\rightarrow} {{j}}\right). \\ $$$$\:\mathrm{Calculate}\:\:\underset{{x}\rightarrow\mathrm{0}^{+} \:} {\mathrm{lim}}\:{f}\left({x}\right),\:\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\:{f}\left({x}\right). \\ $$…
Question Number 85868 by M±th+et£s last updated on 25/Mar/20 $${if}\:{f}\left({x}\right)=\lfloor{x}^{\mathrm{2}} \rfloor\:\: \\ $$$${and}\:{A}=\underset{{x}\rightarrow\mathrm{0}} {{lim}}\left({f}\left({x}\right)−{f}\left(−{x}\right)\right) \\ $$$${and}\:{B}={f}\left({x}\right)+{f}\left(−{x}\right)\:\:{when}\:{x}=\mathrm{0} \\ $$$$ \\ $$$${find}\:{A}\:{and}\:{B} \\ $$ Commented by mr…
Question Number 20138 by tammi last updated on 22/Aug/17 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}−\mathrm{cos}\:{ax}}{\mathrm{1}−\mathrm{cos}\:{bx}} \\ $$$$ \\ $$ Answered by ajfour last updated on 22/Aug/17 $$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{2sin}\:^{\mathrm{2}} \left(\frac{{ax}}{\mathrm{2}}\right)}{\mathrm{2sin}\:^{\mathrm{2}}…
Question Number 85625 by jagoll last updated on 23/Mar/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{x}^{\mathrm{n}} −\left(\mathrm{sin}\:\mathrm{x}\right)^{\mathrm{n}} }{\left(\mathrm{sin}\:\mathrm{x}\right)^{\mathrm{n}+\mathrm{2}} } \\ $$ Answered by john santu last updated on 23/Mar/20 $${by}\:{Maclaurin}\:{series}…
Question Number 151111 by alcohol last updated on 18/Aug/21 Answered by Olaf_Thorendsen last updated on 18/Aug/21 $$\mathrm{On}\:\mathrm{peut}\:\mathrm{eventuellement}\:\mathrm{avoir}\:\mathrm{3}\:\mathrm{balles} \\ $$$$\mathrm{noires}\:\mathrm{au}\:\mathrm{cours}\:\mathrm{des}\:\mathrm{3},\:\mathrm{4},\:\mathrm{5},…\:{k}\:\mathrm{premiers} \\ $$$$\mathrm{tirages}\:\mathrm{mais}\:\mathrm{pour}\:\mathrm{en}\:\mathrm{etre}\:\mathrm{certain},\:\mathrm{il}\:\mathrm{faut} \\ $$$$\mathrm{au}\:\mathrm{moins}\:\mathrm{tirer}\:\mathrm{1003}\:\mathrm{balles}. \\ $$$$\mathrm{Au}\:\mathrm{pire},\:\mathrm{on}\:\mathrm{aurait}\:\mathrm{500}\:\mathrm{balles}\:\mathrm{blanches},…
Question Number 85542 by TawaTawa1 last updated on 22/Mar/20 Commented by TawaTawa1 last updated on 22/Mar/20 $$\mathrm{Evaluate}:\:\:\:\:\:\underset{{x}\rightarrow\frac{\mathrm{x}}{\mathrm{2}}} {\mathrm{lim}}\:\:\left(\mathrm{x}\:\:−\:\:\frac{\pi}{\mathrm{2}}\right)\:\mathrm{tan}\:\mathrm{x} \\ $$ Commented by mathmax by abdo…
Question Number 151001 by EDWIN88 last updated on 17/Aug/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt[{\mathrm{3}}]{\mathrm{2}+\mathrm{sin}\:^{\mathrm{2}} {x}}−\sqrt[{\mathrm{3}}]{\mathrm{1}+\mathrm{cos}\:\mathrm{2}{x}}}{{x}\:\mathrm{tan}\:{x}}\:=?\: \\ $$$$\:\:\: \\ $$ Answered by john_santu last updated on 17/Aug/21…