7
$\begingroup$

Are there examples of functions $f$ such that $\int_0^\infty f\text{d}x$ exists, but $\lim_{x\to\infty}f(x)\neq 0$?

I curious because I know for infinite series, if $a_n\not\to 0$, then $\sum a_n$ diverges. I'm wondering if there is something similar for improper integrals.

  • 6
    $f(x)=\begin{cases} 1 &\text{if } x \in \mathbb Z \\ 0 &\text{otherwise}\end{cases}$?2012-09-16
  • 0
    The classical [Fresnel Integral](http://en.wikipedia.org/wiki/Fresnel_integral)2012-09-16
  • 0
    https://math.stackexchange.com/questions/2527941/if-f-in-l1-bbb-r-dx-then-prove-that-for-almost-every-x-in-bbb-r-lim-lim2018-05-30
  • 0
    https://math.stackexchange.com/questions/2401286/does-this-integral-converge-or-diverge-int-bbb-r-left-frac2-cos-x3?noredirect=1&lq=12018-05-30

3 Answers 3