∵x属于正实数
∴x>0,且(x-1)²≥0 ==>x²+1-2x≥0
==>x²+1≥2x>0
==>1≥2x/(x²+1)>0
==>1/2≥x/(x²+1)>0
故x/(x^2+1)的取值范围是(0,1/2].