Suppose $f$ is a continuous and $f\in L^1[0,\infty)$. I want help in finding examples where
1. if $\lim_{x\to\infty} f(x)$ exists, then $\lim_{x\to\infty} f(x)= 0$.
- if $\lim_{x\to\infty} f(x)$ does not exists, then $\lim_{x\to\infty} f(x)\neq 0$.
Edit: 2. an example in which $f\in L^1[0,\infty)$ and $\lim_{x\to \infty} f(x)$ does not exist.
PS
Could some please find an appropriate title? Thanks.