if we consider 1/exp(sinx) and take the limit as x--> infinity i think it's intuitively clear that the limit doesn't exist because -1<=sinx<=1. But if we multiply top and bottom by x (which doesn't change 1/exp(sinx)) and apply L'Hopital's rule we can argue that the limit is 0. Try differentiating top and bottom and look at the result. there is a factor of x in the denominator which will take the denominator to infinity while the numerator is 1.
can someone resolve this inconsistency?