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Q.
If $f\left(x\right)= \begin{vmatrix}\cos x&x&1\\ 2\sin x&x^{2}&2x\\ \tan x&x&1\end{vmatrix}$ , then $ \displaystyle\lim_{x \to0 } \frac{f' \left(x\right)}{x} $
JEE MainJEE Main 2018Continuity and Differentiability
Solution:
$f ( x )=\begin{vmatrix}\cos x & x & 1 \\2 \sin x & x ^{2} & 2 x \\\tan x & x & 1\end{vmatrix}$
$R _{1}- R _{1}- R _{3} $
$f ( x )=\begin{vmatrix}\cos x -\tan x & 0 & 0 \\2 \sin x & x ^{2} & 2 x \\\tan x & x & 1
\end{vmatrix}$
$=\cos x -\tan x \left[ x ^{2}-2 x ^{2}\right] = x ^{2} \tan x - x ^{2} \cos x$
$f '( x )=2 x \tan x + x ^{2} \sec ^{2} x -2 x \cos x + x ^{2} \sin x $
$\frac{ f '( x )}{ x }=2 \tan x + x \sec ^{2} x -2 \cos x + x \sin x $
$\displaystyle\lim _{x \rightarrow 0} \frac{ f '( x )}{ x }=0+0-2+0=-2$