Let f(x) = begin{vmatrix} cos x & x & 1 2 si | vedclass

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(B) Given $f(x) = \begin{vmatrix} \cos x & x & 1 \ 2 \sin x & x^2 & 2x \ 	an x & x & 1 \end{vmatrix}$.Applying column operation $C_1 	o C_1 - C_3$

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$-2$

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(B) Given $f(x) = \begin{vmatrix} \cos x & x & 1 \ 2 \sin x & x^2 & 2x \ 	an x & x & 1 \end{vmatrix}$.Applying column operation $C_1 	o C_1 - C_3$,we get:$f(x) = \begin{vmatrix} \cos x - 1 & x & 1 \ 2 \sin x - 2x & x^2 & 2x \ 	an x - 1 & x & 1 \end{vmatrix}$.Expanding along the third column or simplifying,we observe that as $x 	o 0$,$f(x) \approx \begin{vmatrix} -x^2/2 & x & 1 \ -2x^3/6 & x^2 & 2x \ x-1 & x & 1 \end{vmatrix}$.More precisely,$f'(x)$ can be found by differentiating the determinant. Using the property of derivative of a determinant:$f'(x) = \begin{vmatrix} -\sin x & 1 & 0 \ 2 \sin x & x^2 & 2x \ 	an x & x & 1 \end{vmatrix} + \begin{vmatrix} \cos x & 1 & 0 \ 2 \cos x & 2x & 2 \ 	an x & x & 1 \end{vmatrix} + \begin{vmatrix} \cos x & x & 1 \ 2 \sin x & x^2 & 2x \ \sec^2 x & 1 & 0 \end{vmatrix}$.Evaluating the limit $\lim_{x 	o 0} \frac{f'(x)}{x}$ using $L$'Hopital's rule or Taylor series expansion,we find the value is $-2$.

Let $f(x) = \begin{vmatrix} \cos x & x & 1 \\ 2 \sin x & x^2 & 2x \\ \tan x & x & 1 \end{vmatrix}$. Then $\lim_{x \to 0} \frac{f'(x)}{x} =$

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