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Author: Tinku Tara

Find-inf-m-n-m-n-N-m-lt-n-2-

Question Number 208103 by depressiveshrek last updated on 05/Jun/24 $$\mathrm{Find}\:\mathrm{inf}\left\{\frac{{m}}{{n}}\:\mid\:{m},\:{n}\:\in\:\mathbb{N},\:{m}<{n}−\mathrm{2}\right\} \\ $$ Answered by A5T last updated on 05/Jun/24 $${Since}\:{m},{n}\in\mathbb{N},\:\frac{{m}}{{n}}>\mathrm{0},{so}\:\mathrm{0}\:{is}\:{a}\:{lower}\:{bound}. \\ $$$$\frac{{m}}{{n}}\geqslant\frac{\mathrm{1}}{{n}}\:{and}\:\frac{\mathrm{1}}{{n}}\:{is}\:{in}\:{the}\:{set}. \\ $$$${For}\:{any}\:\epsilon>\mathrm{0},\:{one}\:{can}\:{always}\:{find}\:{n}\:{such}\:{that} \\…

Question-208097

Question Number 208097 by efronzo1 last updated on 05/Jun/24 $$\:\:\:\:\downharpoonleft\underline{\:} \\ $$ Answered by MM42 last updated on 05/Jun/24 $$\mathrm{2}^{{x}−{m}+\mathrm{1}} −\mathrm{2}^{{x}−{m}} \:=\mathrm{8} \\ $$$$\Rightarrow\mathrm{2}^{{x}−{m}} =\mathrm{8}\Rightarrow{x}−{m}=\mathrm{3}\Rightarrow{x}={m}+\mathrm{3}…

5-x-1-3-x-2-4-lt-0-

Question Number 208076 by hardmath last updated on 04/Jun/24 $$\left(\mathrm{5}\:−\:\mid\mathrm{x}\mid\right)^{−\:\frac{\mathrm{1}}{\mathrm{3}}} \:\left(\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{4}\right)\:<\:\mathrm{0} \\ $$ Answered by TonyCWX08 last updated on 04/Jun/24 $${This}\:{inequality}\:{are}\:{defined}\:{when}\:{x}\in\langle−\mathrm{5},\mathrm{5}\rangle \\ $$$${Two}\:{Possible}\:{Cases} \\…

lim-n-1-n-a-1-n-2-a-2-n-2-a-n-1-n-2-

Question Number 208093 by depressiveshrek last updated on 04/Jun/24 $$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}}{{n}}\left(\left({a}+\frac{\mathrm{1}}{{n}}\right)^{\mathrm{2}} +\left({a}+\frac{\mathrm{2}}{{n}}\right)^{\mathrm{2}} +…+\left({a}+\frac{{n}−\mathrm{1}}{{n}}\right)^{\mathrm{2}} \right) \\ $$ Answered by MM42 last updated on 04/Jun/24 $$={lim}_{{n}\rightarrow\infty} \frac{\mathrm{1}}{{n}}\underset{{i}=\mathrm{1}}…

Find-sin-2x-pi-4-

Question Number 208075 by hardmath last updated on 04/Jun/24 $$\mathrm{Find}:\:\:\:\int\:\mathrm{sin}\:\left(\mathrm{2x}\:−\:\frac{\pi}{\mathrm{4}}\right)\:=\:? \\ $$ Answered by TonyCWX08 last updated on 04/Jun/24 $${sin}\left(\mathrm{2}{x}−\frac{\pi}{\mathrm{4}}\right) \\ $$$$={sin}\left(\mathrm{2}{x}\right){cos}\left(\frac{\pi}{\mathrm{4}}\right)−{sin}\left(\frac{\pi}{\mathrm{4}}\right){cos}\left(\mathrm{2}{x}\right) \\ $$$$=\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\left({sin}\left(\mathrm{2}{x}\right)−{cos}\left(\mathrm{2}{x}\right)\right) \\…