Mathematics - Determinant Question with Solution | TestHub

$\begin{array}{rl}& f(x)=\left|\begin{array}{ccc}\cos x& x& 1\\ 2\sin x& x^2& 2x\\ \tan x& x& 1\end{array}\right|\\ =& \cos x\left(x^2-2x^2\right)-x(2\sin x-2x\tan x)\\ =& -x^2\cos x-2x\sin x+2x^2\tan x\\ +2x\sin x-x^2\tan x)& \\ =& x^2\tan x-x^2\tan x\\ \Rightarrow & f^{\prime }(x)=2x(\tan x-\cos x)+x^2\left({\sec }^{2}x+\sin x\right)\\ \therefore & \underset{x\to 0}{\lim }\frac{f^{\prime }(x)}{x}\\ =& \underset{x\to 0}{\lim }\frac{2x(\tan x-\cos x)+x^2\left({\sec }^{2}x+\sin x\right)}{x}\\ =& \underset{x\to 0}{\lim }(\tan x-\cos x)+x\left({\sec }^{2}x+\sin x\right)\\ =& 2(0-1)+0=-2\\ \text{ So, }\underset{x\to 0}{\lim }\frac{f^{\prime }(x)}{x}=-2& \end{array}$