Question Number 74041 by FCB last updated on 18/Nov/19 Commented by mathmax by abdo last updated on 18/Nov/19 $${let}\:{A}_{{n}} =\left(\mathrm{1}+\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}}\right)^{\frac{\mathrm{1}}{{sin}\left(\pi\sqrt{\mathrm{1}+{n}^{\mathrm{2}} }\right)}} \:\Rightarrow \\ $$$$\left.{ln}\left({A}_{{n}}…
Question Number 139530 by mnjuly1970 last updated on 28/Apr/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:…….{advanced}\:\:{calculus}…… \\ $$$$\:{prove}\:\:{that}:: \\ $$$$\:\:\:{lim}_{{n}\rightarrow\infty} \left\{\frac{\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} {n}^{{n}+\mathrm{1}} }{{n}!}\:\frac{{d}^{\:{n}} }{{dx}^{{n}} }\left(\frac{{ln}\left({x}\right)}{{x}}\right)\mid_{{x}={n}} \right\}=\gamma \\ $$$$\:\gamma\::\:\:\:{euler}\:−{mascheroni}\:{constant} \\ $$$$ \\…
Question Number 73948 by Hardy lanes last updated on 17/Nov/19 $${lim}\:\:\:\left(\frac{\mathrm{sin}\:^{\mathrm{2}} \mathrm{2}{x}}{{x}}\right) \\ $$$${x}\rightarrow\mathrm{0} \\ $$ Commented by mathmax by abdo last updated on 17/Nov/19…
Question Number 73736 by aliesam last updated on 15/Nov/19 $$\underset{{x}\rightarrow\mathrm{0}} {{lim}x}^{{x}} \\ $$ Answered by FCB last updated on 15/Nov/19 Commented by aliesam last updated…
Question Number 73730 by TawaTawa last updated on 15/Nov/19 Answered by mind is power last updated on 15/Nov/19 $${f}\left({x}\right)={e}^{{x}} −{e}^{{c}} −{e}^{{c}} \left({x}−{c}\right) \\ $$$$\Rightarrow{f}'\left({x}\right)={e}^{{x}} −{e}^{{c}}…
Question Number 73712 by FCB last updated on 15/Nov/19 Answered by MJS last updated on 15/Nov/19 $$\underset{{x}\rightarrow\mathrm{0}^{−} } {\mathrm{lim}}\frac{\mathrm{1}}{{x}}\:=−\infty \\ $$$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\frac{\mathrm{1}}{{x}}=+\infty \\ $$$$\Rightarrow\:\underset{{x}\rightarrow\mathrm{0}}…
Question Number 139140 by mathlove last updated on 23/Apr/21 Answered by TheSupreme last updated on 23/Apr/21 $${sin}\left({x}\right)={x}−\frac{{x}^{\mathrm{3}} }{\mathrm{3}}+{o}\left({x}^{\mathrm{3}} \right) \\ $$$${li}\underset{{x}} {{m}}\frac{\mathrm{2019}{x}−\mathrm{2019}{x}+\frac{\mathrm{2019}^{\mathrm{3}} }{\mathrm{3}}{x}^{\mathrm{3}} }{\mathrm{2020}{x}−\mathrm{2020}{x}+\frac{\mathrm{2020}^{\mathrm{3}} }{\mathrm{3}}{x}^{\mathrm{3}}…
Question Number 73572 by Rio Michael last updated on 13/Nov/19 $${determine}\:{wether}\:{or}\:{not}\:{the}\:{function}\:{f},{where} \\ $$$${f}\left({x}\right)\:=\:\begin{cases}{\mathrm{2}{x}\:+\:\mathrm{1},\:\mathrm{0}\leqslant\:{x}\:<\mathrm{2}}\\{\mathrm{7}−{x},\:\:\:\mathrm{2}\:\leqslant\:{x}\:<\:\mathrm{4}}\\{\frac{\mathrm{3}{x}}{\mathrm{4}}\:,\:\:\mathrm{4}\:\leqslant\:{x}\:<\:\mathrm{6}}\end{cases} \\ $$$${is}\:{continuous}\:{in}\:{the}\:{interval}\:\left[\mathrm{0},\mathrm{6}\left[\right.\right. \\ $$ Commented by kaivan.ahmadi last updated on 13/Nov/19 $${lim}_{{x}\rightarrow\mathrm{2}^{−}…
Question Number 73570 by Rio Michael last updated on 13/Nov/19 $${f}\::\:{x}\:\rightarrow\:\begin{cases}{\mathrm{1}\:+\:{x},\:{if}\:{x}<\mathrm{1}}\\{\mathrm{2}{x}−\mathrm{1},{if}\:{x}>\mathrm{1}}\end{cases} \\ $$$${investigate}\:{the}\:{existence}\:{and}\:{non}\:{existence}\:{of}\:{the} \\ $$$${limit}\:{of}\:{f}\:{at}\:{the}\:{point}\:{x}\:=\mathrm{1} \\ $$ Commented by kaivan.ahmadi last updated on 13/Nov/19 $${lim}_{{x}\rightarrow\mathrm{1}^{−}…
Question Number 73566 by Rio Michael last updated on 13/Nov/19 $${show}\:{that}\:{f}\left({x}\right)\:=\:\mid{x}\mid\:{is}\:{not}\:{differentiable}\:{at}\:{x}=\mathrm{0},\:{where}\:\mid{x}\mid \\ $$$${denotes}\:{he}\:{absolute}\:{value}\:{function} \\ $$ Commented by Rio Michael last updated on 13/Nov/19 $${thanks}\:{sir}, \\…