Question Number 220852 by fantastic last updated on 20/May/25 $$\underset{{x}\rightarrow\mathrm{0}} {{Lim}}\left\{\frac{{xe}^{{x}} −{log}\left(\mathrm{1}+{x}\right)}{{x}^{\mathrm{2}} }\right\} \\ $$ Answered by SdC355 last updated on 20/May/25 $$\frac{\frac{\mathrm{d}\:\:}{\mathrm{d}{x}}\left({xe}^{{x}} −\mathrm{ln}\left({x}+\mathrm{1}\right)\right)}{\frac{\mathrm{d}\:}{\mathrm{d}{x}}\:{x}^{\mathrm{2}} }=\frac{\left({x}+\mathrm{1}\right){e}^{{x}}…
Question Number 220843 by Nicholas666 last updated on 20/May/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\alpha\:\in\:\mathbb{R} \\ $$$$\:\:\:\:\:\mathrm{lim}_{{x}\rightarrow\mathrm{1}} \:\frac{\left(\mathrm{1}\:−\:{x}\right)^{\alpha} }{\:^{\mathrm{3}} \sqrt{\mathrm{1}\:−\:{x}^{\mathrm{4}} }}\:\:\:\:\:\:\:\:\in\left(\mathrm{0},\infty\right) \\ $$$$ \\ $$ Commented by SdC355…
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Question Number 220764 by Nicholas666 last updated on 18/May/25 $$ \\ $$$$\:\:\boldsymbol{\mathrm{L}}=\:\boldsymbol{\mathrm{lim}}\underset{\:\boldsymbol{{n}}\rightarrow\infty} {\:}\left(\underset{\boldsymbol{{k}}=\mathrm{1}} {\overset{\boldsymbol{{n}}} {\sum}}\:\frac{\boldsymbol{{k}}}{\boldsymbol{{n}}^{\mathrm{2}} +\boldsymbol{{k}}^{\mathrm{2}} }\right).\left(\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \boldsymbol{{e}}^{−\boldsymbol{{x}}^{\mathrm{2}} } \boldsymbol{{dx}}\overset{−\mathrm{1}} {\right)}.\left(\underset{\boldsymbol{{m}}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{\boldsymbol{{m}}} }{\left(\mathrm{2}\boldsymbol{{m}}+\mathrm{1}\right)\mathrm{3}^{\boldsymbol{{m}}}…
Question Number 220380 by MrGaster last updated on 12/May/25 $$\underset{{n}\rightarrow\infty} {\mathrm{lim}tan}\left[\frac{\pi}{\mathrm{4}}+\frac{\mathrm{1}}{{n}}\right]^{{n}} =? \\ $$ Answered by SdC355 last updated on 12/May/25 $$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\:\mathrm{tan}\left(\left(\frac{\pi}{\mathrm{4}}+\frac{\mathrm{1}}{{n}}\right)^{{n}} \right)=\mathrm{tan}\left(\mathrm{0}\right)=\mathrm{0} \\…
Question Number 219832 by mnjuly1970 last updated on 02/May/25 $$ \\ $$$$\:\:\:\:\:{lim}_{{n}\rightarrow\infty} \:{n}\left(\frac{\mathrm{1}}{\mathrm{1}+{n}}\:+\frac{\mathrm{1}}{\mathrm{2}+{n}}\:+…+\frac{\mathrm{1}}{\mathrm{2}{n}}\:−{ln}\left(\mathrm{2}\right)\right)=? \\ $$$$ \\ $$ Answered by universe last updated on 03/May/25 Commented…
Question Number 219731 by Nescio last updated on 01/May/25 Answered by breniam last updated on 03/May/25 $$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }×\frac{{x}}{\mathrm{log}\left(\mathrm{1}+\mathrm{2}{x}\right)}×\mathrm{tan}\left({x}\right)={L} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }=\mathrm{1}…
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Question Number 219223 by fantastic last updated on 20/Apr/25 Commented by Nicholas666 last updated on 22/Apr/25 https://bekicotsspace.quora.com/How-do-I-solve-the-problem-Let-math-f-x-lim_-n-to-infty-left-frac-n-n-x-n-x-frac-n-2-cdots-x-frac-n?ch=10&oid=220368951&share=05ee32a3&srid=5Xg5SU&target_type=post Answered by zetamaths last updated on 20/Apr/25 $$\frac{{f}'\left(\mathrm{3}\right)}{{f}\left(\mathrm{3}\right)}\geqslant\frac{{f}'\left(\mathrm{2}\right)}{{f}\left(\mathrm{2}\right)}\:\:\:…
Question Number 218703 by Nicholas666 last updated on 14/Apr/25 $$ \\ $$$$\:\:\:\:{Prove};\:{lim}_{{x}\rightarrow\mathrm{0}} \:\frac{{x}\:−\:{sin}\:{x}}{{x}^{\mathrm{3}} }\:=\:\frac{\mathrm{1}}{\mathrm{6}}\:\: \\ $$$$ \\ $$ Answered by SdC355 last updated on 14/Apr/25…