Question Number 8964 by Joel575 last updated on 08/Nov/16 $$\mathrm{Can}\:\mathrm{you}\:\mathrm{solve}\:\mathrm{this}\:\mathrm{problem}? \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{log}\:\left({x}^{\mathrm{3}} \:+\:\left(\mathrm{log}\:{x}\right)^{\mathrm{3}} \right)}{\mathrm{log}\:\left({x}^{\mathrm{2}} \:+\:\left(\mathrm{log}\:{x}\right)^{\mathrm{2}} \right)}\: \\ $$ Commented by sou1618 last updated on…
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Question Number 140000 by bramlexs22 last updated on 03/May/21 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{1}+\mathrm{x}.\mathrm{2}^{\mathrm{x}} }{\mathrm{1}+\mathrm{x}.\mathrm{3}^{\mathrm{x}} }\right)^{\frac{\mathrm{2}}{\mathrm{x}^{\mathrm{2}} }} =? \\ $$ Commented by MJS_new last updated on 03/May/21 $$\mathrm{the}\:\mathrm{answer}\:\mathrm{should}\:\mathrm{be}\:\frac{\mathrm{4}}{\mathrm{9}}\:\mathrm{I}\:\mathrm{think}…
Question Number 74383 by aliesam last updated on 23/Nov/19 Commented by mathmax by abdo last updated on 23/Nov/19 $${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…