If $\int_a^{+\infty} f(x)dx $ converges, and $\lim_{x\to + \infty} f(x) = L$. Then prove that $L=0$

Question

I have the following statement which I have to prove.

If $\int_a^{+\infty} f(x)dx $ converges, and $\lim_{x\to + \infty} f(x) = L$. Then prove that $L=0$

The idea is to prove it only using the improper integral definition, but I don’t know how to do it.
Thanks in advance!

Edit: I wrote down the limit definition for f, and then integrate both sides. Doing that I got:
$\int_a^{+\infty} L-\varepsilon \ dx \leq L’ $
(Being $L’= \int_a^{+\infty} f(x)dx $ (Since it converges)
That is the hint that the professor showed us in a drawing. But how can I use this information to conclude that L must be equal to 0?

Answer 1

Hint: Assume that $L \neq 0$, and prove that the integral diverges.