The given limit $ = \mathop {\lim }\limits_{x \to 0} \frac{{{{\left( {2\sin \frac{x}{2}\cos \frac{x}{2} – 2\sin \frac{x}{2}} \right)}^2} + {{\left( {2{{\sin }^2}\frac{x}{2}} \right)}^3}}}{{2{{\sin }^2}x\cos x – 8\cos x{{\sin }^2}\frac{x}{2} – \frac{4}{3}{{\sin }^4}x}}$

$ = \mathop {\lim }\limits_{x \to 0} \frac{{4{{\sin }^2}\frac{x}{2}{{\left( {\cos \frac{x}{2} – 1} \right)}^2} + 8{{\sin }^6}\frac{x}{2}}}{{8{{\sin }^2}\frac{x}{2}{{\cos }^2}\frac{x}{2}\cos x – 8\cos x{{\sin }^2}\frac{x}{2} – \frac{4}{3}.16{{\sin }^4}\frac{x}{2}{{\cos }^4}\frac{x}{2}}}$

$ = \mathop {\lim }\limits_{x \to 0} \frac{{4{{\sin }^2}\frac{x}{2}.4{{\sin }^4}\frac{x}{4} + 8{{\sin }^6}\frac{x}{2}}}{{8{{\sin }^2}\frac{x}{2}\cos x\left( {{{\cos }^2}\frac{x}{2} – 1} \right) – \frac{{64}}{3}{{\sin }^4}\frac{x}{2}{{\cos }^4}\frac{x}{2}}}$

$ = \mathop {\lim }\limits_{x \to 0} \frac{{8{{\sin }^2}\frac{x}{2}\left( {2{{\sin }^4}\frac{x}{4} + {{\sin }^4}\frac{x}{2}} \right)}}{{ – 8{{\sin }^2}\frac{x}{2}\cos x{{\sin }^2}\frac{x}{2} – \frac{{64}}{3}{{\sin }^4}\frac{x}{2}{{\cos }^4}\frac{x}{2}}}$

$ = \mathop {\lim }\limits_{x \to 0} \frac{{8{{\sin }^2}\frac{x}{2}\left( {2{{\sin }^4}\frac{x}{4} + 16{{\sin }^4}\frac{x}{4}{{\cos }^4}\frac{x}{4}} \right)}}{{ – 8{{\sin }^4}\frac{x}{2}\cos x – \frac{{64}}{3}{{\sin }^4}\frac{x}{2}{{\cos }^4}\frac{x}{2}}}$

$ = \mathop {\lim }\limits_{x \to 0} \frac{{16{{\sin }^2}\frac{x}{2}{{\sin }^4}\frac{x}{4}\left( {1 + 8{{\cos }^4}\frac{x}{4}} \right)}}{{ – 8{{\sin }^4}\frac{x}{2}\left( {\cos x + \frac{8}{3}{{\cos }^4}\frac{x}{2}} \right)}}$

$ = \mathop {\lim }\limits_{x \to 0} \frac{{ – 2{{\sin }^4}\frac{x}{4}\left( {1 + 8{{\cos }^4}\frac{x}{4}} \right)}}{{4{{\sin }^2}\frac{x}{4}{{\cos }^2}\frac{x}{4}\left( {\cos x + \frac{8}{3}{{\cos }^4}\frac{x}{2}} \right)}}$

$ = \mathop {\lim }\limits_{x \to 0} \frac{{ – {{\tan }^2}\frac{x}{4}\left( {1 + 8{{\cos }^4}\frac{x}{4}} \right)}}{{2\left( {\cos x + \frac{8}{3}{{\cos }^4}\frac{x}{2}} \right)}} = 0$