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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 $\lim_{x\to\infty} f(x)= 0$.

  1. 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.

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    I forgot the function $f(x)$.2012-03-22
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    I don't understand 2. Something (the limit $\lim_x f(x)$) which doesn't exist, can't be $\ne 0$.2012-03-22
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    @martini Sorry. I edited it.2012-03-22

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