Question

For what values of s will the limit of x approach infininty for lnx/x^s.

Answer 1

Your question is unclear. Do you want the limit of lnx/x^s AS x-> infinity to be finite? or infinite?

I suggest you use L'Hopital's rule to compute the limit. That involves takng the ratio of the derivatives of the numerator and the denominator, which is (1/x)/[sx^(s-1)]= (1/s)[x^-s)]
That limit will be zero if s>0 and infinite if s<0.