Let $~f~$ be a non-negative continuous function on $~\mathbb{R}~$ such that $~\int_{0}^{∞} f(x)~dx~$ is finite,then $~\lim_{x\to ∞} f(x) = 0~$.

Question

Let $~f~$ be a non-negative continuous function on $~\mathbb{R}~$ such that $$~\int_{0}^{∞} f(x)~dx~$$ is finite,then $$~\lim_{x\to ∞} f(x) = 0~.$$

Is this statement true? I am trying to find a counter example. And I don't know how it is false.

"Is there any counter example?": Did you bother to look at the duplicate questions people found for you? — David C. Ullrich, Sep 14 '19 at 10:38

score 0 · Answer 1 · answered Sep 14 '19 at 08:23

Let $f$ such that for all $n\in\mathbb{N}^*$, the graph of f on [n,n+1] draws a triangular spike of area $\frac{1}{n^2}$ and which base is $\frac{1}{n^3}$ (hence the hedge of this triangle is something like $n$). You have that $$\int_0^{+\infty}f(x)dx=\zeta(2)$$ but the statement $\lim\limits_{x\rightarrow +\infty}f(x)=0$ is not satisfied.

Let $~f~$ be a non-negative continuous function on $~\mathbb{R}~$ such that $~\int_{0}^{∞} f(x)~dx~$ is finite,then $~\lim_{x\to ∞} f(x) = 0~$.

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