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Question-192927

Question Number 192927 by 073 last updated on 31/May/23 Answered by aba last updated on 31/May/23 $$\mathrm{n}!\left(\mathrm{n}+\mathrm{1}\right)^{\mathrm{m}} \approx\sqrt{\mathrm{2}\pi\mathrm{n}}\left(\frac{\mathrm{n}}{\mathrm{e}}\right)^{\mathrm{n}} ×\mathrm{n}^{\mathrm{m}} \:\:\wedge\:\left(\mathrm{n}+\mathrm{m}\right)!\approx\sqrt{\mathrm{2}\pi\left(\mathrm{n}+\mathrm{m}\right)}\left(\frac{\mathrm{n}+\mathrm{m}}{\mathrm{e}}\right)^{\mathrm{n}+\mathrm{m}} \\ $$$$\Rightarrow\frac{\mathrm{n}!\left(\mathrm{n}+\mathrm{1}\right)^{\mathrm{m}} }{\left(\mathrm{n}+\mathrm{m}\right)!}\approx\frac{\sqrt{\mathrm{2}\pi\mathrm{n}}}{\:\sqrt{\mathrm{2}\pi\left(\mathrm{n}+\mathrm{m}\right)}}×\frac{\mathrm{n}^{\mathrm{n}+\mathrm{m}} }{\mathrm{e}^{\mathrm{n}} }×\frac{\mathrm{e}^{\mathrm{n}+\mathrm{m}}…

lim-a-x-1-b-x-1-c-x-1-a-b-c-1-x-x-0-

Question Number 127358 by Fareed last updated on 29/Dec/20 $$ \\ $$$$\mathrm{lim}\left(\frac{\mathrm{a}^{\mathrm{x}+\mathrm{1}} +\mathrm{b}^{\mathrm{x}+\mathrm{1}} +\mathrm{c}^{\mathrm{x}+\mathrm{1}} }{\mathrm{a}+\mathrm{b}+\mathrm{c}}\right)^{\frac{\mathrm{1}}{\mathrm{x}}} =? \\ $$$$\mathrm{x}\Rightarrow\mathrm{0} \\ $$ Answered by Ar Brandon last…

Question-192884

Question Number 192884 by 073 last updated on 30/May/23 Commented by witcher3 last updated on 03/Jun/23 $$\mathrm{let}\:\mathrm{u}=\mathrm{x}+\mathrm{y},\mathrm{v}=\mathrm{x}−\mathrm{y} \\ $$$$\mathrm{g}\left(\mathrm{u},\mathrm{v}\right)=\left(\mathrm{x},\mathrm{y}\right) \\ $$$$\mathrm{J}_{\mathrm{g}} =\begin{pmatrix}{\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{2}}}\\{\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\:−\frac{\mathrm{1}}{\mathrm{2}}}\end{pmatrix} \\ $$$$\begin{pmatrix}{\mathrm{x}}\\{\mathrm{y}}\end{pmatrix}=\frac{\mathrm{1}}{\mathrm{2}}\begin{pmatrix}{\mathrm{u}+\mathrm{v}}\\{\mathrm{u}−\mathrm{v}}\end{pmatrix} \\…