lim-x-x-xlnx-

Question Number 6189 by enigmeyou last updated on 17/Jun/16

\lim_{x \to + \infty} ⌊ x^{⌊ x l n x ⌋} ⌋ = ?

Commented by FilupSmith last updated on 18/Jun/16

⌊ x ⌋ is only really applicable for non

integers . ⌊ x \ln x ⌋ is only applicable if

x \ln (x) is not integer answer . \infty can, in a sense

be use as an^{'} {integer}^{'} as ⌊ \infty ⌋ = \infty

That is, \lim_{x \to \infty} ⌊ x^{⌊ x \ln x ⌋} ⌋ = \lim_{x \to \infty} x^{x \ln x}

= \lim_{x \to \infty} x^{\ln (x^{x})}

= \infty^{\ln (\infty)}

= \infty

Commented by FilupSmith last updated on 18/Jun/16

\lim_{x \to \infty} ⌊ x^{⌊ x \ln (x) ⌋} ⌋ = ⌊ \lim_{x \to \infty} x^{⌊ \ln (\lim_{x \to \infty} x^{x}) ⌋} ⌋

= ⌊ \infty^{⌊ \ln (\infty^{\infty}) ⌋} ⌋

= ⌊ \infty^{⌊ \infty^{\infty} ⌋} ⌋

= ⌊ \infty^{⌊ \infty ⌋} ⌋

= ⌊ \infty ⌋

= \infty