Let $\mathcal{C}=\left\{f: \mathbb{R} \rightarrow \mathbb{R} \mid f\right.$ is differentiable, and $\left.\lim _{x \rightarrow \infty}\left(2 f(x)+f^{\prime}(x)\right)=0\right\}$ Which of the following statements is/are correct?
(a) For each $L$ with $0 \neq L<\infty$, there exists $f \in \mathcal{C}$ such that $\lim _{x \rightarrow \infty} f(x)=L$.
(b) For all $f \in \mathcal{C}, \lim _{x \rightarrow \infty} f(x)=0$.
(c) There exists $f \in \mathcal{C}$ such that $\lim _{x \rightarrow \infty} f(x)$ does not exist.
(d) There exists $f \in \mathcal{C}$ such that $\lim _{x \rightarrow \infty} f(x)=\frac{1}{2}$.
I can see the case of $e^{-2x}$ which is not useful to deal the options. Can we confirm the limit of $f$ is $0$ at infinity?
Can I connect the result:
'$\lim _{x \rightarrow \infty}(f(x)+f^{\prime}(x))=L \in (0,\infty),$ implies $\lim _{x \rightarrow \infty} f(x)=L$ provided the existence' ?
