Finding a absolutely integrable function $f$ such that $\lim_{x\to\infty}f(x)\neq 0$

Question

Find a function $f : \Bbb R \rightarrow \Bbb R $ such that $\int_\Bbb R |f(x)|dx < \infty $ but for x $ \rightarrow \infty$, $f(x)\not\rightarrow 0$.

Is there a way of generating such functions? I have just tried various functions, but have been unable to find any that satisfy both conditions.

Thanks in advance.

Answer 1

One such example is $\sum_{n=1}^{\infty}n\cdot \chi_{(n,n+1/n^3)}.$

Answer 2

For $n\in \Bbb N$ let $f_n(x)=1-n^2|x-n|$, and let $$f(x)=\max\{0,\max\{\,f_n(x)\mid n\in\Bbb N\,\} \}$$ Then $f$ is continuous, $\int_{\Bbb R}|f(x)|\mathrm dx\le\sum \frac 1{n^2}=\frac{\pi^2}6$, and $f(x)\not\to 0$ as $x\to\infty$

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With $f_n(x)=n\cdot(1-n^3|x-n|)$, the resulting $f$ is additionally unbounded.

Answer 3

Consider the sequence $f_n=e^{\frac{-x}{n}}sin(x^2)$. The pointwise limit is $sin(x^2)$ and $\int_{\mathbb{R}}sin(x^2)dx=\sqrt{\frac{\pi}{2}}$. Note this function is continuous on all of $\mathbb{R}$.