Behavior of Improper Integral

Question

I am trying to understand better the behavior of improper integrals depending on the function.

I think that this items are correct by intuition, but I can't seem to find a theorem or lemma that proves them.

1. If $f$ is a continuous function and $\int_a^\infty f$ $dx = L$ (a finite number), then $\lim_{x \to \infty} f(x) = 0$

I think that this makes sense if $f$ is positive. In order for the graph not to define an infinite large area, the function must get close to the $x$ axis after some $x_0$ and remain there. The same perhaps could be said if $f$ is decreasing.

And the opposite of 1:

2. If $\lim_{x \to \infty} f(x) = 0$, then $\int_a^\infty f$ $dx = L \lt \infty$

This would only hold if $f$ is positive.

Am I on the right track? How could I prove that both items, if they are correct, are true?

Thanks a lot!

Answer 1

For the first one, consider $f(x)$ is a triangle with height $1$ on $[n,n+\frac{1}{n^2}]$ and vanishes otherwise. If $a>0$, the integral $\int_a^\infty f(x) dx\le \sum \frac{1}{2n^2}<\infty$. But $f\not\to 0$ as $x\to \infty$.

The second is also wrong. For a counterexample, $f(x)=\frac{1}{x^a}, 0<a\le 1$.

Answer 2

Hint

With respect to the second one consider $f(x)=\frac{1}{x}.$