Mathematics - Determinant Question with Solution | TestHub

Given $f\left(x\right)=\left|\begin{array}{ccc}\cos x& x& 1\\ 2\sin x& x^2& 2x\\ \tan x& x& 1\end{array}\right|$

$=\cos x\left(x^2-2x^2\right)-2\sin x\left(x-x\right)+\tan x\left(2x^2-x^2\right)$

$=-x^2\cos x+\tan x.x^2$

$=x^2\left(\tan x-\cos x\right)$

$\Rightarrow \underset{x\to 0}{\mathrm{Lim}}\frac{f^{\text{'}}\left(x\right)}{x}=\underset{x\to 0}{\mathrm{Lim}}\frac{2x\left(\tan x-\cos x\right)+x^2\left({\sec }^{2}x+\sin x\right)}{x}$

$=\underset{x\to 0}{\mathrm{Lim}}2\left(\tan x-\cos x\right)+x\left({\sec }^{2}x+\sin x\right)$

$=-2$