Let $ f:[1,+\infty)\to[0,+\infty)$ be a function such that $ \int_{1}^{+\infty}f(x)dx$ is convergent. I have 3 questions as follows:

question(1) Does $ f(x)$ have limit at infinity?

question(2) If the answer of question (1) is yes, Is $ \lim_{x\to\infty}f(x)=0$ true?

question(3) If the answers of questions (1) and (2) are NEGATIVE, What can we say about their answers, if $ f$ is continuous function?