Given, f\left( x \right) = \left| {\matrix{ {\cos x} & x & 1 \cr {2\sin x} & {{x^2}} & {2x} \cr {\tan x} & x & 1 \cr } } \right| = cosx(x 2 - 2x 2 ) - x(2 sinx - 2x tanx) + (2x sinx - x 2 tanx) = x 2 (tanx - cosx) $\therefore$ ${f^{'}}(x)$ = 2x (tanx - cosx) + x 2 (sec 2 x + sinx) $\therefore$ $\mathop {\lim }\limits_{x \to 0} {{f'\left( x \right)} \over x}$ = $\mathop {\lim }\limits_{x \to o} {{2x(\tan x - \cos x) + {x^2}({{\sec }^2}x + \sin x)} \over x}$ = $\mathop {\lim }\limits_{x \to o} \,\,2(\tan x - \cos x) + x({\sec ^2}x + \sin x)$ = 2 (0-1) + 0 = -2