$f: [a. \infty ) \rightarrow \mathbb R$ is differentiable and $\int_a^\infty f(t)dt $ and $\int_a^\infty f^{\prime}(t)dt $ are convergent then $f(t) \rightarrow 0$ as $t \rightarrow \infty$
my answer :
$\int_a^x f^{\prime}(t)dt =f(x) - f(a)$
$\int_a^\infty f^{\prime}(t)dt $ is convergent $\implies$ $\lim_{x \rightarrow \infty} f(x) = l $(where $l$ is finite ).to show $l=0$.
if $l \neq 0 \implies \exists