Limits Continuity And Differentiability

Suppose \(\mathop {\lim }\limits_{x \to 0} {{F(x)} \over {{x^3}}}\) exists and is equal to L, where

\(F(x) = \left| {\matrix{ {a + \sin {x \over 2}} & { - b\cos x} & 0 \cr { - b\cos x} & 0 & {a + \sin {x \over 2}} \cr 0 & {a + \sin {x \over 2}} & { - b\cos x} \cr } } \right|\).

Then, \(-\)112 L is equal to ___________.