TITLE: $L=\lim_{x\to\infty}(f(x)+f'(x))$ exists . Which of the following statements is\are correct?
QUESTION [10 upvotes]: Let $f$ be a continously differentiable function on $\mathbb R$.  Suppose that 
$$L=\lim_{x\to\infty}(f(x)+f'(x))$$ exists.  If $0<L<\infty$, then which of the following statements is\are correct?



If $\lim_{x\to\infty} f'(x)$ exists, then it is $0$.
If $\lim_{x\to\infty} f(x)$ exists, then it is $L$.
If $\lim_{x\to\infty} f'(x)$ exists, then $\lim_{x\to\infty}f(x)=0$.
If $\lim_{x\to\infty} f(x)$ exists, then $\lim_{x\to\infty}f'(x)=0$.



My Guess
I could not conclude the answer and prove that properly. But, I guess that it must be 1 and 2. help me.

REPLY [12 votes]: Hint: Check this: $$\lim_{x \to +\infty} f(x) = \lim_{x \to +\infty}\frac{e^xf(x)}{e^x} \color{red}{=} \lim_{x \to +\infty}\frac{e^x(f(x)+f'(x))}{e^x} = \lim_{x \to +\infty}f(x)+f'(x) = L,$$ by $\color{red}{\text{L'Hospital's rule}}$.