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  4. If f(x)=| sin x cos x tan x x3 x2 x 2 x 1 x| then undersetx arrow 0 textLim (f(x)/x2) is equal to

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Q. If $f(x)=\begin{vmatrix}\sin x & \cos x & \tan x \\ x^3 & x^2 & x \\ 2 x & 1 & x\end{vmatrix}$ then $\underset{x \rightarrow 0}{\text{Lim}} \frac{f(x)}{x^2}$ is equal to

Determinants

Solution:

Given limit $=\underset{x \rightarrow 0}{\text{Lim}}\begin{vmatrix}\frac{\sin x}{x} & \cos x & \frac{\tan x}{x} \\ \frac{x^3}{x} & x^2 & \frac{x}{x} \\ \frac{2 x}{x} & 1 & \frac{x}{x}\end{vmatrix}=\begin{vmatrix}1 & 1 & 1 \\ 0 & 0 & 1 \\ 2 & 1 & 1\end{vmatrix}$
$=1(0-1)+2(1-0)=1$.