Question

$f\left(x\right)=\left|\begin{array}{ccc}\cos x& x& 1\\ 2\sin x& x^2& 2x\\ \tan x& x& 1\end{array}\right|$. The value of $\underset{x\to 0}{lim}\frac{f\left(x\right)}{x}$ is equal to

• A. $1$
• B. $-1$
• C. $0$
• D. None of these

Answer 1

Answer: (C)

$0$

$f\left(x\right)=\left|\begin{array}{ccc}\cos x& x& 1\\ 2\sin x& x^2& 2x\\ \tan x& x& 1\end{array}\right|$
Taking $x$ common from $C_2$
${lim}_{x\to 0}\frac{f\left(x\right)}{x}={lim}_{x\to 0}\left|\begin{array}{ccc}\cos x& 1& 1\\ 2\sin x& x& 2x\\ \tan x& 1& 1\end{array}\right|=\left|\begin{array}{ccc}1& 1& 1\\ 0& 0& 0\\ 0& 1& 1\end{array}\right|=0$
Hence, option $C$ is correct.