Answer
See proof
Work Step by Step
Given
$$\lim _{x \rightarrow \infty} f(x)=0$$
Since
\begin{aligned}
\int_0^{\infty} f^{\prime}(x) d x=\lim _{t \rightarrow \infty} \int_0^t f^{\prime}(x) d x,
\end{aligned}
by the Fundamental Theorem of Calculus:
\begin{aligned}
\lim _{t \rightarrow \infty} \int_0^t f^{\prime}(x) d x&=\lim _{t \rightarrow \infty}[f(t)-f(0)]\\
&=\lim _{t \rightarrow \infty} f(t)-f(0)\\
&=0-f(0)\\
&=-f(0)
\end{aligned}
