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Author: Tinku Tara

can-a-function-f-be-such-that-for-a-R-f-X-a-R-a-a-X-X-R-R-gt-0-lim-x-a-f-x-lim-x-a-f-x-lt-

Question Number 1093 by 123456 last updated on 12/Jun/15 $$\mathrm{can}\:\mathrm{a}\:\mathrm{function}\:{f}\:\mathrm{be}\:\mathrm{such}\:\mathrm{that}\:\mathrm{for}\:{a}\in\mathbb{R} \\ $$$${f}:\mathrm{X}/\left\{{a}\right\}\rightarrow\mathbb{R},\left({a}−\delta,{a}+\delta\right)\in\mathrm{X},\mathrm{X}\subset\mathbb{R},\delta\in\mathbb{R},\delta>\mathrm{0} \\ $$$$\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:{f}\left({x}\right)=\infty \\ $$$$\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:\mid{f}'\left({x}\right)\mid<\infty \\ $$ Answered by prakash jain last…

n-1-sin-n-n-2-

Question Number 132162 by Dwaipayan Shikari last updated on 11/Feb/21 $$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{sin}\left({n}\right)}{{n}^{\mathrm{2}} } \\ $$ Answered by mnjuly1970 last updated on 11/Feb/21 $$\frac{\mathrm{1}}{\mathrm{2}{i}}\left[{li}_{\mathrm{2}} \left({e}^{{i}}…

I-0-pi-x-pi-x-sin-x-dx-

Question Number 1085 by 123456 last updated on 10/Jun/15 $$\mathrm{I}=\underset{\mathrm{0}} {\overset{\pi} {\int}}\frac{{x}\left(\pi−{x}\right)}{\mathrm{sin}\:{x}}{dx} \\ $$ Commented by 123456 last updated on 10/Jun/15 $${f}\left({x}\right)=\frac{{x}\left(\pi−{x}\right)}{\mathrm{sin}\:{x}} \\ $$$${f}\left(\mathrm{0}^{+} \right)\overset{?}…