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

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

Solution:

$f(x)=\sin x\left(x^{2}-x\right)-\cos x\left(x^{3}-2 x^{2}\right)+\tan x\left(x^{3}-2 x^{3}\right)$
$\underset{_{x \rightarrow 0}}{ lt}\frac{f(x)}{x^{2}}= \underset{_{x \rightarrow 0}}{lt}\left(\sin x-\frac{\sin x}{x}-(x-2) \cos x-x \tan x\right)$
$=0-1-(0-2) x 1-0=-1+2=1$