Let $\,\,f:\mathbb{R}\rightarrow\mathbb{R}\,\,$ be a continuous function such that $ \ \int_{0}^{\infty}{f(x)} dx $ exists. Which of the following statements are always true ?
1.if $\lim_{n\rightarrow\infty}f(x)$ exists, then $\lim_{n\rightarrow\infty}f(x)=0$
2.$\lim_{n\rightarrow\infty}f(x)$ exists and is $0$
3.in case $f$ is nonnegative function ,$\lim_{n\rightarrow\infty}f(x)$ must exist and zero
4.in case of $f$ is differentiable function , $\lim_{n\rightarrow\infty}$f$'$(x) must exist and is zero
I think 1 is always true , no idea for other three
please help