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

Answer 1

#0#

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#

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Answer 2

Emily Ammerman

To find the limit of ln(ln(x))/x as x approaches infinity, we can use L'Hôpital's rule. Taking the derivative of the numerator and denominator separately, we get (1/ln(x))/1. As x approaches infinity, ln(x) also approaches infinity, so the limit becomes 1/infinity, which is equal to 0. Therefore, the limit of ln(ln(x))/x as x approaches infinity is 0.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.