$L=\lim_{x\to\infty}(f(x)+f'(x))$ exists . Which of the following statements is\are correct? 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.

Hint:
If $\lim f'(x) = M$, then $\lim f(x) = L-M$
Use the MVT: $f(x+1) - f(x) = f'(\xi)$ with $x < \xi < x+1$.  
