Let
$C = \{ f:\mathbb{R}\rightarrow \mathbb{R}| f$ is differentiable, and $\lim_{x\rightarrow \infty }\left ( 2f\left ( x \right ) +f{}’\left ( x \right )\right )=0\left \} \right.$.
Which of the following statements is correct?
- For each $L$ with $0\neq L< \infty$, there exists $f\in C$ such that $\lim_{x\rightarrow \infty }f\left ( x \right )=L$
- For all $f \in C, \lim_{x\rightarrow \infty }f\left ( x \right )=0$
- There exists $f \in C$ such that $\lim_{x\rightarrow \infty }f\left ( x \right )$ does not exist
- there exists $f \in C$ such that $\lim_{x\rightarrow \infty }f\left ( x \right )\frac{1}{2}$
