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Question: If .\(f(x) = \left| \begin{matrix} \sin x & \cos x & \tan x \\ x^{3} & x^{2} & x \\ 2x & 1 & 1 \end{...

If .f(x)=∣sin⁡xcos⁡xtan⁡xx3x2x2x11∣f(x) = \left| \begin{matrix} \sin x & \cos x & \tan x \\ x^{3} & x^{2} & x \\ 2x & 1 & 1 \end{matrix} \right|, then lim⁡x→0f(x)x2\lim_{x \rightarrow 0}\frac{f(x)}{x^{2}} is

A

3

B

–1

C

0

D

1

Answer

1

Explanation

Solution

f(x)=x(x−1)sin⁡x−(x3−2x2)cos⁡x−x3tan⁡xf(x) = x(x - 1)\sin x - (x^{3} - 2x^{2})\cos x - x^{3}\tan x

=x2sin⁡x−x3cos⁡x−x3tan⁡x+2x2cos⁡x−xsin⁡x= x^{2}\sin x - x^{3}\cos x - x^{3}\tan x + 2x^{2}\cos x - x\sin xHence, lim⁡x→0f(x)x2=lim⁡x→0(sin⁡x−xcos⁡x−xtan⁡x+2cos⁡x− sin⁡xx) =0−0−0+2−1=1\lim_{x \rightarrow 0}\frac{f(x)}{x^{2}} = \lim_{x \rightarrow 0}\left( \sin x - x\cos x - x\tan x + 2\cos x - \left. \ \frac{\sin x}{x} \right) \right.\ = 0 - 0 - 0 + 2 - 1 = 1.