Category: Limits

Question Number 102020 by Dwaipayan Shikari last updated on 06/Jul/20 $$\underset{x\to 0}{\lim }{\left(tan(\frac{\pi }{4}-x)\right)}^{\frac{1}{x}}$$ Answered by john santu last updated on 06/Jul/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\mathrm{1}+\left(\mathrm{tan}\:\left(\frac{\pi}{\mathrm{4}}−{x}\right)−\mathrm{1}\right)\right)^{\frac{\mathrm{1}}{{x}}} \…

Question Number 101977 by Ar Brandon last updated on 05/Jul/20 $$\mathrm{Given}2\mathrm{functions},\mathrm{f}\mathrm{and}\mathrm{g},\mathrm{n}-\mathrm{times}\mathrm{derivable}\mathrm{within}$$$$\mathrm{the}\mathrm{open}\mathrm{interval},\mathbb{R}\mathrm{and}\mathrm{verify}\mathrm{the}\mathrm{property}$$$$\mathrm{f}\left({\mathrm{x}}_0\right)={\mathrm{f}}^{\left(\mathrm{k}\right)}\left({\mathrm{x}}_0\right)=0,\mathrm{g}\left({\mathrm{x}}_0\right)={\mathrm{g}}^{\left(\mathrm{k}\right)}\left({\mathrm{x}}_0\right)=0,\mathrm{\forall }\mathrm{k}\in \{1,2,\dots ,\mathrm{n}-1\}$$$$\mathrm{Show}\:\mathrm{that}\:\underset{\mathrm{x}\rightarrow\mathrm{x}_{\mathrm{0}} } {\mathrm{lim}}\frac{\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{g}\left(\mathrm{x}\right)}=\frac{\mathrm{f}^{\left(\mathrm{n}\right)}…

Question Number 167421 by infinityaction last updated on 16/Mar/22 Answered by mindispower last updated on 16/Mar/22 $${cot}^{\mathrm{2}} \left(\frac{{k}\pi}{\mathrm{2}{n}+\mathrm{1}}\right)={cot}^{\mathrm{2}} \left({s}\pi\right)=\left({i}\frac{{e}^{{is}} +{e}^{−{is}} }{{e}^{{is}} −{e}^{−{is}} }\right)^{\mathrm{2}} = \…

Question Number 167350 by infinityaction last updated on 13/Mar/22 Answered by Jamshidbek last updated on 13/Mar/22 $$\mathrm{Telegram}:@\mathrm{math}\mathrm{\_}\mathrm{undergraduate}$$$$\mathrm{Problem}15.$$$$\mathrm{This}\mathrm{has}\mathrm{problem}\mathrm{solution}\mathrm{in}\mathrm{telegram}\mathrm{channel}$$ Commented by…

Question Number 167295 by mnjuly1970 last updated on 12/Mar/22 Answered by qaz last updated on 12/Mar/22 $$\underset{\mathrm{n}\to \mathrm{\infty }}{\lim }{\left(\frac{\left(2\mathrm{n}\right)!}{{\mathrm{n}}^{\mathrm{n}}\mathrm{n}!}\right)}^{1/\mathrm{n}}$$$$=\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\frac{\left(\mathrm{2n}\right)!}{\mathrm{n}^{\mathrm{n}} \mathrm{n}!}\centerdot\frac{\left(\mathrm{n}−\mathrm{1}\right)^{\mathrm{n}−\mathrm{1}} \left(\mathrm{n}−\mathrm{1}\right)!}{\left(\mathrm{2n}−\mathrm{2}\right)!} \…