Why does lim_(x->oo) x-x^(((1)/(x))^(1)/(x))-(log^2) x=0?

Bruce Sherman

Bruce Sherman

Answered question

2022-10-09

Why does lim x x x 1 x 1 x log 2 x = 0 ?
Moreover, why is
x x 1 x 1 x log 2 x ?

Answer & Explanation

Gabriella Hensley

Gabriella Hensley

Beginner2022-10-10Added 6 answers

By a Taylor expansion,
( 1 x ) 1 / x = exp ( 1 x ln x ) = 1 ln x x + O ( ln 2 x x 2 ) .
Thus the second term equals
( 1 + δ ) exp ( ln x ( 1 ln x x ) ) = x ( 1 ln 2 x x + O ( ln 4 x x 2 ) ) ( 1 + δ )
with δ = O ( ln 3 x / x 2 ), so this gives the claim.

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