Question

If $f\left(x\right)=\left|\begin{array}{ccc}\cos x& x& 1\\ 2\sin x& x^2& 2x\\ \tan x& x& 1\end{array}\right|$ , then $\underset{x\to 0}{\lim }\frac{f^{\prime }\left(x\right)}{x}$

• A. does not exist.
• B. exists and is equal to 2.
• C. exists and is equal to 0.
• D. exists and is equal to -2.

Answer 1

Answer: (D)

$f(x)=\left|\begin{array}{ccc}\cos x& x& 1\\ 2\sin x& x^2& 2x\\ \tan x& x& 1\end{array}\right|$
$R_1-R_1-R_3$
$f(x)=\left|\begin{array}{ccc}\cos x-\tan x& 0& 0\\ 2\sin x& x^2& 2x\\ \tan x& x& 1<br/><br/>\end{array}\right|$
$=\cos x-\tan x\left[x^2-2x^2\right]=x^2\tan x-x^2\cos x$
$f^{\prime }(x)=2x\tan x+x^2{\sec }^{2}x-2x\cos x+x^2\sin x$
$\frac{f^{\prime }(x)}{x}=2\tan x+x{\sec }^{2}x-2\cos x+x\sin x$
$\underset{x\to 0}{\lim }\frac{f^{\prime }(x)}{x}=0+0-2+0=-2$