Question Number 74383 by aliesam last updated on 23/Nov/19 Commented by mathmax by abdo last updated on 23/Nov/19 $${A}\left({x}\right)=\frac{\mathrm{16}\sqrt{{x}−\sqrt{{x}}}−\mathrm{3}\sqrt{\mathrm{2}}{x}−\mathrm{4}\sqrt{\mathrm{2}}}{\mathrm{16}\left({x}−\mathrm{4}\right)^{\mathrm{2}} }{let}\:{use}\:{hospital}\:{theorem}\:\:{let}\:{f}\left({x}\right)=\mathrm{16}\sqrt{{x}−\sqrt{{x}}}\:−\mathrm{3}\sqrt{\mathrm{2}}{x}−\mathrm{4}\sqrt{\mathrm{2}} \\ $$$${and}\:{g}\left({x}\right)=\mathrm{16}\left({x}−\mathrm{4}\right)^{\mathrm{2}} \:\:{we}\:{have}\:{f}^{'} \left({x}\right)=\mathrm{16}\frac{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}\sqrt{{x}}}}{\mathrm{2}\sqrt{{x}−\sqrt{{x}}}}\:−\mathrm{3}\sqrt{\mathrm{2}} \\…
Question Number 8839 by tawakalitu last updated on 31/Oct/16 $$\mathrm{Prove}\:\mathrm{that}.\: \\ $$$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\mathrm{1}\:+\:\mathrm{n}\right)^{\mathrm{1}/\mathrm{n}} \:=\:\mathrm{e} \\ $$ Answered by FilupSmith last updated on 31/Oct/16 $${L}=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\mathrm{1}\:+\:{n}\right)^{\mathrm{1}/{n}}…
Question Number 8628 by sou1618 last updated on 18/Oct/16 $${is}\:{it}\:{always}\:{satisfying}? \\ $$$$\boldsymbol{{A}}=\mathrm{lim}\left[{n}\rightarrow\infty\right]\int{f}\left({n},{x}\right){dx} \\ $$$$\boldsymbol{{B}}=\int\mathrm{lim}\left[{n}\rightarrow\infty\right]{f}\left({n},{x}\right){dx} \\ $$$${A}={B}?? \\ $$$${please}\:{show}\:{counter}\:{example} \\ $$$$ \\ $$$$ \\ $$$${checking} \\…
Question Number 74041 by FCB last updated on 18/Nov/19 Commented by mathmax by abdo last updated on 18/Nov/19 $${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}}…