The given limit = ${e^{\mathop {\lim }\limits_{x \to 0} \frac{{\tan x – \sin x}}{{{{\sin }^3}x}}}}$

$ = {e^{\mathop {\lim }\limits_{x \to 0} \frac{{\frac{1}{{\cos x}} – 1}}{{{{\sin }^2}x}}}}$

$ = {e^{\mathop {\lim }\limits_{x \to 0} \frac{{1 – \cos x}}{{\cos x.{{\sin }^2}x}}}}$

$ = {e^{\mathop {\lim }\limits_{x \to 0} \frac{{1 – \cos x}}{{\cos x.(1 – {{\cos }^2}x)}}}}$

$ = {e^{\mathop {\lim }\limits_{x \to 0} \frac{1}{{\cos x.(1 + \cos x)}}}}$

$ = {e^{1/2}} = \sqrt e $