Question

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

• A. 0
• B. -1
• C. -2
• D. 2

Answer 1

The correct option is A 0

$f\left(x\right)=\left|\begin{array}{ccc}cosx& x& 1\\ 2sinx& x^2& 2x\\ tanx& x& 1\end{array}\right|$
$R_3\to R_3{-R}_1$
$f\left(x\right)=\left|\begin{array}{ccc}cosx& x& 1\\ 2sinx& x^2& 2x\\ \tan x-\cos x& 0& 0\end{array}\right|$
$=(\tan x-\cos x)x^2$
Now, ${lim}_{x\to 0}f\left(x\right)$
$={lim}_{x\to 0}(\tan x-\cos x)x^2$
$=0$