TITLE: If $f(x) + f'(x) + f''(x) \to A$ as $x \to \infty$, then show that $f(x) \to A$ as $x \to \infty$
QUESTION [16 upvotes]: This problem is an extension to the simpler problem which deals with $f(x) + f'(x) \to A$ as $x \to \infty$ (see problem 2 on my blog).

If $f$ is twice continuously differentiable in some interval $(a, \infty)$ and $f(x) + f'(x) + f''(x) \to A$ as $x \to \infty$ then show that $f(x) \to A$ as $x \to \infty$.

However, the approach based on considering sign of $f'(x)$ for large $x$ (which applies to the simpler problem in the blog) does not seem to apply here. Any hints on this problem?
I believe that a similar generalization concerning expression $\sum\limits_{k = 0}^{n}f^{(k)}(x) \to A$ is also true, but I don't have a clue to prove the general result.

REPLY [5 votes]: The idea in Jochen's answer provides the following:
Generalization. Let $p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$, and assume that all the roots of $p$ have negative real parts (which implies that $a_0\ne 0$). If $f: (b,\infty)\to\mathbb C$ is $n$ times differentiable and
$$
\lim_{t\to\infty} f^{(n)}(t)+a_{n-1}f^{(n-1)}(t)+\cdots+a_0 f(t)=A,
$$ 
then 
$$
\lim_{t\to\infty}f(t)=\frac{A}{a_0}.
$$