$$ \begin{aligned} \lim_{x\rightarrow 0} \frac {tanx -sinx} {x^3} \frac {sinx}{cosx} -sinx \\ \frac {sinx}{cosx} - \frac {sin^2x}{sinx} \\ \frac { sin^2x -cosx * sin^2x} {sinx * cosx} \\ \frac {sinx(1-cosx)} {cosx} \\ \frac {sinx(1-cosx)} {cosx * x^3} \\ \frac {\frac{x^3}{x}}{cosx ~ x^3} \\ \frac {x^3}{2} ~\frac{1}{cosx * x^3} \\ \frac {1}{2cosx} \\

\lim_{x\rightarrow 0} \frac {tanx -sinx} {x^3}=\frac {1}{2} \end{aligned}

$$