General — Medium JEE math MCQ
Let $f:(0, \infty) \rightarrow \mathbb{R}$ be a differentiable function such that $f^{\prime}(x)=2-\frac{f(x)}{x}$ for all $x \in(0, \infty)$ and $f(1) \neq 1$. Then
- A. $\lim _{x \rightarrow 0+} f^{\prime}\left(\frac{1}{x}\right)=1$
- B. $\lim _{x \rightarrow 0+} x f\left(\frac{1}{x}\right)=2$
- C. $\lim _{x \rightarrow 0+} x^{2} f^{\prime}(x)=0$
- D. $|f(x)| \leq 2$ for all $x \in(0,2)$
Solution
The correct option is **A**.