TITLE: Limit of a function. Integration problem
QUESTION [9 upvotes]: I have this problem:
Let $c\in \mathbb{R}$. If $\int_c^\infty f(x)dx$ converges, then $$\lim_{x\to \infty} f(x)$$ exist and is $0$. Moreover, if $f$ is monotonic, $\lim_{x\to \infty} xf(x)$ exist and is equal to $0$.
Although I can't assume that $f$ is continuous, I have tihis question:
If $F$ is a function with continuous first derivative, such that $$\lim_{x\to \infty}F(x)=L,$$ for some $L\in\mathbb{R}$. Is it true that $$\lim_{x\to \infty} F'(x)=0\;?$$

REPLY [8 votes]: It's true iff $\rm\ lim\ F\,'$ exists. Andre gave a counterexample if $\rm\ lim\ F\,'$ does not exist. Conversely: 
Theorem $\ $ If $\rm\,\ F + F\,'\!\to L\ $ as $\rm\ x\to\infty\ $ then $\rm\ F\to L,\  F\,'\!\to 0,\,\ $ by this L'Hôpital slick trick:
$$\rm \lim_{x\to\infty}\ F(x)\ =\ \lim_{x\to\infty}\frac{e^x\, F(x)}{e^x}\ =\ \lim_{x\to\infty}\frac{e^x\, (F(x)+F\:'(x))}{e^x}\ =\ \lim_{x\to\infty}\ (F(x)+F\:'(x)) $$
The above employs a slightly generalized form of L'Hospital's rule mentioned here.
This folklore L'Hospital trick is somewhat notorious due to the fact that the problem appeared in Hardy's classic calculus texbook A Course of Pure Mathematics, but with a less elegant solution. For example, see Landau; Jones: $\:$ A Hardy Old Problem, $\:$ Math. Magazine 56 (1983) 230-232. Below is a table of the various possibilities, with examples, where FTE = Fails To Exist.