Integral convergence and limit 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.
 A: Hint: What would happen to $\int_1^\infty{|f(x)|dx}$ if $\lim_{x\rightarrow \infty}{f(x)}=L$ is non-zero?
A: 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, $\forall \epsilon > 0 \ \exists x^* > 1 : \forall x > x^* f(x) \in (L-\epsilon, L+\epsilon)$. 
A: 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
