Question

How I resolve this limit?

`lim_(n->+oo)(1/logn)^((log(logn))/n)`

THANKS!

Answer 1

`1`

Explanation:

`lim_(n->+oo)(1/ln n)^((ln ln n)/n)`

`lim_(n->+oo)e^ { ln( (1/ln n)^((ln ln n)/n) )}`

`= exp( lim_(n->+oo) ((ln ln n)/n) ln( 1/ln n) )`

`= exp( lim_(n->+oo) -(ln ln n)^2/n)`

`= exp( 0 )`

`= 1`

I wrote `exp(x)` instead of `e^x` because the latter doesn't look very good with the entire limit in the exponent. `exp(x)=e^x`

In the limit of a very slowly increasing `(ln ln n)^2` in the numerator versus a must faster increasing `n` in the denominator, the denominator wins and the limit is zero.