Question Number 113357 by mohammad17 last updated on 12/Sep/20
$$\left.{lim}_{{x}\rightarrow\infty} \left({ln}\left({ln}\left({lnx}\right)\right)\right)\right)^{\frac{\mathrm{1}}{{x}}} \\ $$
Commented by Aziztisffola last updated on 12/Sep/20
$$\left.{lim}_{{x}\rightarrow\infty} \left({ln}\left({ln}\left({lnx}\right)\right)\right)\right)^{\frac{\mathrm{1}}{{x}}} =\mathrm{1} \\ $$
Commented by mohammad17 last updated on 12/Sep/20
$${can}\:{you}\:{give}\:{me}\:{the}\:{details}\:{please}\:{sir} \\ $$
Commented by Aziztisffola last updated on 12/Sep/20
$$\left.\:\mathrm{L}=\mathrm{li}\underset{{x}\rightarrow\infty} {\mathrm{m}}\left({ln}\left({ln}\left({lnx}\right)\right)\right)\right)^{\frac{\mathrm{1}}{{x}}} \\ $$$$\mathrm{lnL}=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\left.{ln}\left({ln}\left({lnx}\right)\right)\right)}{{x}}=\mathrm{0} \\ $$$$\:\mathrm{L}=\mathrm{e}^{\mathrm{0}\:} \Rightarrow\:\mathrm{L}=\mathrm{1} \\ $$
Commented by mohammad17 last updated on 12/Sep/20
$${help}\:{me}\:{sir} \\ $$
Commented by mohammad17 last updated on 12/Sep/20
$$ \\ $$$${lim}_{{x}\rightarrow\infty} \left(\frac{\sqrt{{x}}}{{log}\sqrt{{x}}}\right) \\ $$
Commented by mohammad17 last updated on 12/Sep/20
$${help}\:{me}\:{please} \\ $$
Commented by Aziztisffola last updated on 13/Sep/20
$$\mathrm{let}\:\mathrm{t}=\sqrt{\mathrm{x}}\Rightarrow\mathrm{x}\rightarrow\infty\Rightarrow\mathrm{t}\rightarrow\infty \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\sqrt{{x}}}{\mathrm{ln}\left({x}\right)}=\underset{{t}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{t}}{\mathrm{ln}\left(\mathrm{t}\right)}=\underset{\mathrm{t}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}}{\frac{\mathrm{ln}\left(\mathrm{t}\right)}{\mathrm{t}}}=\infty \\ $$$$\left(\underset{\mathrm{t}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{ln}\left(\mathrm{t}\right)}{\mathrm{t}}=\mathrm{0}\right) \\ $$$$\: \\ $$