Question Number 6189 by enigmeyou last updated on 17/Jun/16
$$\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\lfloor{x}^{\lfloor{xlnx}\rfloor} \rfloor=? \\ $$
Commented by FilupSmith last updated on 18/Jun/16
$$\lfloor{x}\rfloor\:\mathrm{is}\:\mathrm{only}\:\mathrm{really}\:\mathrm{applicable}\:\mathrm{for}\:\mathrm{non} \\ $$$$\mathrm{integers}.\:\lfloor{x}\mathrm{ln}{x}\rfloor\:\mathrm{is}\:\mathrm{only}\:\mathrm{applicable}\:\mathrm{if} \\ $$$${x}\mathrm{ln}\left({x}\right)\:\mathrm{is}\:\mathrm{not}\:\mathrm{integer}\:\mathrm{answer}.\:\infty\:\mathrm{can},\mathrm{in}\:\mathrm{a}\:\mathrm{sense} \\ $$$$\mathrm{be}\:\mathrm{use}\:\mathrm{as}\:\mathrm{an}\:'\mathrm{integer}'\:\mathrm{as}\:\lfloor\infty\rfloor=\infty \\ $$$$\mathrm{That}\:\mathrm{is},\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\lfloor{x}^{\lfloor{x}\mathrm{ln}\:{x}\rfloor} \rfloor=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{x}^{{x}\mathrm{ln}\:{x}} \\ $$$$=\underset{{x}\rightarrow\infty} {\mathrm{lim}}{x}^{\mathrm{ln}\left({x}^{{x}} \right)} \\ $$$$=\infty^{\mathrm{ln}\left(\infty\right)} \\ $$$$=\infty \\ $$
Commented by FilupSmith last updated on 18/Jun/16
$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\lfloor{x}^{\lfloor{x}\mathrm{ln}\left({x}\right)\rfloor} \rfloor=\lfloor\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{x}^{\lfloor\mathrm{ln}\left(\underset{{x}\rightarrow\infty} {\mathrm{lim}}{x}^{{x}} \right)\rfloor} \rfloor \\ $$$$=\lfloor\infty^{\lfloor\mathrm{ln}\left(\infty^{\infty} \right)\rfloor} \rfloor \\ $$$$=\lfloor\infty^{\lfloor\infty^{\infty} \rfloor} \rfloor \\ $$$$=\lfloor\infty^{\lfloor\infty\rfloor} \rfloor \\ $$$$=\lfloor\infty\rfloor \\ $$$$=\infty \\ $$