[Math] Given a continuous function which it’s improper integral converges, prove/disprove $\lim_{x\to \infty}f(x)=0$

Question

Prove/Disprove:

$(1)$ If $f$ is a continuous function such that $f > 0, \forall x \in [0,\infty)$ and $\int_0^\infty f $ converges, then $\lim_{x\to \infty}f(x)=0.$

$(2)$ If $f$ is a continuous and monotonic decreasing function such that $f \geq 0, \forall x \in [0,\infty)$ and $\int_0^\infty f $ converges, then $\lim_{x\to \infty}f(x)=0.$

I think $(1)$ is false, but I couldn't find a counter example.

I think $(2)$ is true, tried to prove with Cauchy's criterion for improper integrals and the definition of the limit, but got stuck.

Any help is appreciated.

Answer 1

Hint for (1): tall thin triangles.

(2) $\int_0^\infty f(x)\; dx \ge x f(x)$.