Question

How do you find the Limit of `ln(ln(x))/x` as x approaches infinity?

Answer 1

`0`

Explanation:

You can tell just by inspection that the limit will be zero for the simple reason that log(x) grows more slowly than x, and here it is actually log(log(x)) in the numerator.

You can also nail it more formally with L'Hopital's Rules, as it is `oo/oo` indeterminate

`lim_(x to oo) ln(ln(x))/x`

`= lim_(x to oo) ( 1/ ln(x)* 1/x )/1`

`= lim_(x to oo) 1/ ln(x) * lim_(x to oo) 1/x = 0`