limit x->0 xsin1/x

When limit as x-->0, Put the value of x as zero, as it approaches zero,then the expression becomes, 0 * (a finite value lying between -1 and 1)because sin(1/0) is also a finite value lying between -1 and 1.so, 0 * this finite value gives zero itself. Therefore the value of x sin(1/x) is zero.