a) $\mathop {\lim }\limits_{x \to 0}\frac{tanx-sinx}{x^3}$
=$\mathop {\lim }\limits_{x \to 0}\frac{\frac{sinx-sinx.cosx}{cosx}}{x^3}$
=$\mathop {\lim }\limits_{x \to 0}\frac{sinx(\frac{1-cosx}{cosx})}{x^3}$
=$\mathop {\lim }\limits_{x \to 0}\frac{sinx.(2\frac{sin^2\frac{x}{2}}{cosx})}{x^3}$
=$\mathop {\lim }\limits_{x \to 0}\frac{x.\frac{sinx}{x}.\frac{2}{cosx}(\frac{x}{2})^2.(\frac{sin\frac{x}{2}}{\frac{x}{2}})^2}{x^3}$
=$\mathop {\lim }\limits_{x \to 0}\frac{x^3\frac{sinx}{x}.\frac{1}{2.cosx}.(\frac{sin\frac{x}{2}}{\frac{x}{2}})^2}{x^3}$
=$\frac{1}{2cosx}=\frac{1}{2}$