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 $