One thing seems to be sure: if your sequence converges monotonically to zero (from above, since we're given $\,x_n>0\,$) then any subsequence will bound it elementwise from below: $\,\,x'_n\leq x_n\,,\,\forall n\,$, and then any possible limit of the quotient of both will have to be in $\,[0,1]\,$ , so if we have $\,1 we'll have to begin with a seq. that converges to zero non-monotonically, and this already rules out lots of pretty simple and basic examples, and also shows us that either we put some conditions on the sequence $\,\{x_n\}\,$ or else the answer to your question is : no, not any real $\,x\,$ can be gotten as a limit of that quotient for any sequence.