Suppose $f : \Bbb R \longrightarrow \Bbb R$ be a continuous function such that $\int\limits_{0}^{\infty} f(x)\ dx$ exists finitely. If $\lim\limits_{x \rightarrow \infty} f(x)$ also exists finitely then is it necessarily true that $\lim\limits_{x \rightarrow \infty} f(x) = 0$?

Any suggestion regarding this will be highly appreciated. Thank you very much for your valuable time.

  • 2
    $\begingroup$ If $f$ tends to nonzero, it will be nonzero over an interval of infinite measure. $\endgroup$
    – user65203
    Dec 4, 2019 at 9:50

1 Answer 1


If $L=\lim_{x \to \infty} f(x) >0$ then there exists $M$ such that $f(x) >\frac L 2$ for $x \geq M$ and $\int_M^{\infty} f(x)dx =\infty$. Similarly $L <0$ leads to a contradiction, so $L=0$.

  • 1
    $\begingroup$ Nice answer.+1. $\endgroup$ Dec 4, 2019 at 9:49

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