Lebesgue integrable function vanishing at infinity

Question

I have seen in many places that if a Lebesgue integrable function over $\mathbb R^+$ say, and if $f$ has a finite limit at infinity, then this limit must be zero! I just couldn't find any proof and hope this not too obvious.

Thanks.

Math

Answer 1

Think of area under the curve. If the function approaches a finite, positive limit, can there be a finite area under the curve?

More formally: if $f(x)\to L$ as $x\to\infty$, then for sufficiently large $x$ (say, $x>M$), we must have that $\lvert f(x)-L\rvert<\frac{L}{2}$. In other words, we know that $$ \frac{L}{2}\leq f(x)\leq \frac{3L}{2}\text{ for all }x>M. $$ Now, if $f$ is Lebesgue integrable, you can write $$ \int\limits_{\mathbb{R}^+}f\,dm=\int\limits_{[0,M]}f\,dm+\int\limits_{(M,\infty)}f\,dm $$ and both of these integrals must be finite. But we see that $$ \int\limits_{(M,\infty)}f\,dm\geq\int\limits_{(M,\infty)}\frac{L}{2}\,dm, $$ which is clearly infinite (as $\frac{L}{2}>0$).