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tan-a-b-y-tan-a-b-x-tan2a-

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Question Number 116009 by Khalmohmmad last updated on 30/Sep/20 $$\begin{cases}{\mathrm{tan}\left(\mathrm{a}+\mathrm{b}\right)=\mathrm{y}}\\{\mathrm{tan}\:\left(\mathrm{a}−\mathrm{b}\right)=\mathrm{x}}\end{cases}\:\:\:\mathrm{tan2a}=? \\ $$ Answered by bemath last updated on 30/Sep/20 $$\:\begin{cases}{{a}+{b}={r}}\\{{a}−{b}={s}}\end{cases}\Rightarrow\mathrm{2}{a}\:=\:{r}+{s} \\ $$$$\mathrm{tan}\:\left(\mathrm{2}{a}\right)\:=\:\frac{\mathrm{tan}\:{r}+\mathrm{tan}\:{s}}{\mathrm{1}−\mathrm{tan}\:{r}.\:\mathrm{tan}\:{s}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{{x}+{y}}{\mathrm{1}−{xy}} \\…

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lim-x-x-x-x-x-

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Question Number 116007 by Khalmohmmad last updated on 30/Sep/20 $$\underset{{x}\rightarrow\propto} {\mathrm{lim}}\frac{\sqrt{\mathrm{x}}}{\:\sqrt{\mathrm{x}+\sqrt{\mathrm{x}\sqrt{\mathrm{x}}}}} \\ $$ Answered by bemath last updated on 30/Sep/20 $$=\:\sqrt{\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{{x}}{{x}+\sqrt{{x}\sqrt{{x}}}}}\:=\sqrt{\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{{x}}{{x}+\sqrt{{x}^{\mathrm{2}} \left(\frac{\sqrt{{x}}}{{x}}\right)}}}\: \\…

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x-y-z-R-2x-3y-4z-1-1-x-1-y-1-z-smallest-integer-value-

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Question Number 115951 by Fikret last updated on 29/Sep/20 $${x},{y},{z}\:\epsilon\:{R}^{+} \:\: \\ $$$$\mathrm{2}{x}+\mathrm{3}{y}+\mathrm{4}{z}=\mathrm{1}\:\:\Rightarrow\:\frac{\mathrm{1}}{{x}}+\frac{\mathrm{1}}{{y}}+\frac{\mathrm{1}}{{z}}\:{smallest}\:{integer}\:{value}?\: \\ $$ Answered by 1549442205PVT last updated on 30/Sep/20 $$\mathrm{From}\:\mathrm{the}\:\mathrm{hypothesis}\:\mathrm{we}\:\mathrm{have}\: \\ $$$$\mathrm{P}=\frac{\mathrm{1}}{{x}}+\frac{\mathrm{1}}{{y}}+\frac{\mathrm{1}}{{z}}\:=\frac{\mathrm{2}{x}+\mathrm{3}{y}+\mathrm{4}{z}}{\mathrm{x}}+\frac{\mathrm{2}{x}+\mathrm{3}{y}+\mathrm{4}{z}}{\mathrm{y}}…

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Given-that-the-sum-of-infinity-of-geometric-series-a-2ar-4ar-2-8ar-3-a-2r-n-1-is-3-and-the-sum-of-infinity-of-geometric-series-a-ar-ar-2-ar-3-ar-n-1-is-k-find-the-range-of-

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Question Number 115946 by ZiYangLee last updated on 29/Sep/20 $$\mathrm{Given}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{infinity}\:\mathrm{of}\:\mathrm{geometric} \\ $$$$\mathrm{series}\:{a}−\mathrm{2}{ar}+\mathrm{4}{ar}^{\mathrm{2}} −\mathrm{8}{ar}^{\mathrm{3}} +…{a}\left(−\mathrm{2}{r}\right)^{{n}−\mathrm{1}} \:+…\mathrm{is}\:\mathrm{3} \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{infinity}\:\mathrm{of}\:\mathrm{geometric}\:\mathrm{series} \\ $$$${a}+{ar}+{ar}^{\mathrm{2}} +{ar}^{\mathrm{3}} +…{ar}^{{n}−\mathrm{1}} +…\:\mathrm{is}\:{k},\: \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:{k}. \\…

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