Question

$\mathrm{If}f\left(x\right)=\left(\begin{array}{ccc}x& \sin x& \cos x\\ x^2& \tan x& -x^3\\ 2x& \sin 2x& 5x\end{array}\right|,\mathrm{then}\underset{x\to 0}{\lim }\frac{f\text{'}\left(x\right)}{x}\mathrm{equals}$

• A. 1
• B. 4
• C. 3
• D. -4

Answer 1

The correct option is B

4

$\underset{x\to 0}{\lim }\frac{f\text{'}\left(x\right)}{x}=\underset{x\to 0}{\lim }\frac{f\text{'}\text{'}\left(x\right)}{1}=f\text{'}\text{'}\left(0\right),\mathrm{Now},f\left(x\right)=\left(\begin{array}{ccc}x& \sin x& \cos x\\ x^2& \tan x& -x^3\\ 2x& \sin 2x& 5x\end{array}\right|\mathrm{Upon}\mathrm{differentiating}fw.r.tx\mathrm{twice}\mathrm{and}\mathrm{substituting}x=0,f\text{'}\text{'}\left(0\right)=4\Rightarrow \underset{x\to 0}{\lim }\frac{f\text{'}\left(x\right)}{x}=f\text{'}\text{'}\left(0\right)=4$