\lim_{x\rightarrow 0}\frac{\tan(x)-\sin(x)}{x^3}=\lim_{x\rightarrow 0}\frac{\sin(x)-\sin(x)\cos(x)}{x^3\cos(x)}=\lim_{x\rightarrow 0}\frac{\sin(x)(1-\cos(x))}{x*x^2\cos(x)}=1/2\lim_{x\rightarrow 0}\frac{1}{cos(x)}=\frac 12