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Author: Tinku Tara

lim-z-J-1-z-Y-0-z-lim-z-0-z-1-1-2-z-2-cos-z-1-z-2-z-4-lim-z-0-J-z-h-Y-z-J-z-Y-z-h-

Question Number 219792 by SdC355 last updated on 02/May/25 $$\underset{{z}\rightarrow\infty} {\mathrm{lim}}\:\frac{{J}_{\mathrm{1}} \left({z}\right)}{{Y}_{\mathrm{0}} \left({z}\right)} \\ $$$$\underset{{z}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{z}\frac{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}{z}^{\mathrm{2}} −\mathrm{cos}\left(\frac{{z}}{\mathrm{1}−{z}^{\mathrm{2}} }\right)}{{z}^{\mathrm{4}} } \\ $$$$\underset{{z}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{J}_{\nu} \left({z}+{h}\right){Y}_{\nu} \left({z}\right)−{J}_{\nu} \left({z}\right){Y}_{\nu}…

If-0-lt-a-b-Then-prove-that-a-b-sinx-2sin-2-x-1-sin-2-x-1-sin-2-x-dx-b-a-2-

Question Number 219853 by hardmath last updated on 02/May/25 $$\mathrm{If}\:\:\:\mathrm{0}<\mathrm{a}\leqslant\mathrm{b} \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \left(\mathrm{sinx}\right)^{\mathrm{2}\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:\centerdot\:\left(\mathrm{1}−\mathrm{sin}^{\mathrm{2}} \mathrm{x}\right)^{\mathrm{1}−\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:\mathrm{dx}\:\geqslant\:\frac{\mathrm{b}−\mathrm{a}}{\mathrm{2}} \\ $$ Answered by…