Integral convergence and limit question

Question

I have a question that has come up in a group homework project that neither I nor my partner have any idea how to solve.

I'm hoping someone can give me a hint or some guidance as to how to go about solving it.

Here is the question:

"Suppose that $ \int_1^\infty |f(x)|dx$ converges and $\lim \limits_{x \to \infty}$ $f(x)=L$. What is the value of $L$? Justify your answer."

I have tried a few things but nothing seems to work. I thought I maybe missed some theorem that my professor gave us, like the Dominated Convergence Theorem or the Limit Comparison Theorem, but I don't believe those apply in any way.

I know that if $ \int_1^\infty |f(x)|dx$ converges then so does $ \int_1^\infty f(x)dx$, maybe that could be of some help?

I'm just not even sure how much you can even say about $L$, I mean obviously we won't be able to assign an actual number to it.

Thanks in advance for any help.

Answer 1

Short answer $L=0$; if not, the integral will diverge.

Intuition Assume $f(x)\equiv L$. Then $\int_1^X f(x)dx = (X-1)L \rightarrow \pm \infty$ if $X \rightarrow \infty$, the sign of $\infty$ is the same as the sign of $L$.

Hint Existence of limit implies that, $\forall \epsilon > 0 \ \exists x^* > 1 : \forall x > x^* f(x) \in (L-\epsilon, L+\epsilon)$.

Answer 2

Hint: What would happen to $\int_1^\infty{|f(x)|dx}$ if $\lim_{x\rightarrow \infty}{f(x)}=L$ is non-zero?

Answer 3

Another elegant solution is to assume your limit is some number away from 0, then $\lim_{x\to \infty}|f(x)|=|L|>0$. Then for some $x\in [1,\infty]$, we have for all $y>x$, $||f(y)|-|L||<|L|/2$, or in other words, the absolute value of the function is strictly bounded away from 0 by $|L/2|$. Now you can break up the integral of your function to the regions $[1,y]$ and $[y,\infty]$. Since we are integrating a non-negative function, the part from $[1,y]$ is non-negative. And the part from $[y,\infty]$ is greater than or equal to integrating the constant function $|L/2|$, which is unbounded, a contradiction.

Hence the limit has to be 0