lim[x→0]{(tanx-sinx)/x^3} = lim[x→0] sin(x){1-cos(x)}/{x^3*cos(x)}
=lim[x→0] 2sin(x/2)cos(x/2)*{1-(1-2sin^2(x/2))}/{x^3*cos(x)}
=lim[x→0] 4sin^3(x/2)cos(x/2)/{x^3*cos(x)}
=lim[x→0] {4sin^3(x/2)/x^3}*{cos(x/2)/cos(x)}
=lim[x→0] (4/2^3)*{sin(x/2)/(x/2)}^3*{cos(x/2)/cos(x)}=(1/2)*(1^3)*1=1/2