I don't have any idea about how to solve the following GRE question. Could anyone give me some tips how to solve this kind of problems?
Let $f: (1,\infty) \to [0,\infty)$ be a function, such that the improper integral $\int_1^{\infty} f(x) dx$ converges. Which of the following is true:
I) $\lim_{x \to \infty} f(x)$ exists.
II) If $f(x)$ is monotonous decreasing, then $\lim_{x \to \infty} f(x)$ exists.
III) If $f(x)$ has derivatives of every order then $\lim_{x\to\infty} f(x)$ exists.
Thanks for your help.