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

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? – Subhajit Saha Sep 14 '19 at 8:07
• "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

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.