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Prove: $f(x)$ continuous function and $\int_a^\infty |f(x)|\;dx \lt \infty$ so $\lim_{ x \to \infty } f(x)=0$
Assume $f(x)$ positive and continuous,for improper integrals of the first kind, is there an analog of the $n$-th term divergence test for series ? It would say: $$ \int_{a}^{\infty} f(x)dx\ \hspace{5pt}\mbox{converges} \Rightarrow \lim_{x \to +\infty} f(x) = 0. $$ Give a proof or counterexample.
I think $$ \lim_{x \to +\infty} f(x) \neq 0$$ but I can not give a counterexample.
