I'm looking for an example of f(x) which is absolutely integrable but its limit at positive infinity exists and is not zero.
$$\int_{-\infty}^{\infty}|f(x)|dx< \infty$$ $$\lim_{x \to \infty} f(x)= L\neq 0$$
I know if f'(x) is continuous then the limit at infinity is zero. And there are examples using series where the limit does not exist.
Without considering the existence of f'(x), Is there such function that satisfies the conditions above?
If there is no such function how can I prove it?
There's no such functions: $|f(x)| > \frac{L}2$ for $x > x_0$ and $\int_{\mathbb{R}} | f(x)|dx \ge \int_{x>x_0} | f(x)|dx \ge \int_{x>x_0} L dx = \infty$.
There are integrable functions, such that $\overline{lim}_{x \to \infty} f(x) dx = L > 0$. E.g. $f(x) = \sum_{n \ge 1} I\{ n \le x \le n+2^{-n} \}$.