Question

If $f\left(x\right)=\left|\begin{array}{ccc}\sin x& \cos x& \tan x\\ x^3& x^2& x\\ 2x& 1& x\end{array}\right|$ then $\underset{x\to 0}{lim}\frac{f\left(x\right)}{x^2}=$

• A. 0
• B. 3
• C. 2
• D. 1

Answer 1

The correct option is A 0

$f\left(x\right)=\left|\begin{array}{ccc}\sin x& \cos x& \tan x\\ x^3& x^2& x\\ 2x& 1& x\end{array}\right|$

Expand the determinant along the first row,

$\Rightarrow f\left(x\right)=\sin x(x^3-2x)-\cos x(x^4-2x^2)+\tan x(x^3-2x^3)$

$\Rightarrow \frac{f\left(x\right)}{x^2}=\frac{\sin x}{x}(x^2-2)-\cos x(x^2-2)-x\tan x$

$\therefore \underset{x\to 0}{lim}\frac{f\left(x\right)}{x^2}=\underset{x\to 0}{lim}\frac{\sin x}{x}(x^2-2)-\underset{x\to 0}{lim}\cos x(x^2-2)-\underset{x\to 0}{lim}x\tan x$

$=1(0-2)-1(0-2)-0=0$