lim-x-lnx-x-lnx-x-

Question Number 72495 by Tony Lin last updated on 29/Oct/19

\lim_{x \to \infty} {(\frac{l n x}{x})}^{\frac{l n x}{x}} = ?

Commented by mathmax by abdo last updated on 29/Oct/19

l e t f (x) = {(\frac{l n x}{x})}^{\frac{l n x}{x}} v h a n g e m e n t l n x = t g i v e

f (x) = g (t) = {(\frac{t}{e^{t}})}^{\frac{t}{e^{t}}} = {({t e}^{- t})}^{{t e}^{- t}} x \to + \infty \Leftrightarrow t \to + \infty

g (t) = e^{{t e}^{- t} l n (t e^{- t})} = e^{t e^{- t} (l n t - t)} = e^{t l n (t) e^{- t}} e^{- t^{2} e^{- t}}

b u t {l i m}_{t \to + \infty} t l n (t) e^{- t} = 0 a n d {l i m}_{t \to + \infty} e^{- t^{2} e^{- t}} = 1 \Rightarrow

{l i m}_{x \to + \infty} f (x) = 1

Commented by Tony Lin last updated on 30/Oct/19

t h a n k s s i r

Commented by mathmax by abdo last updated on 30/Oct/19

y o u a r e w e l c o m e .