Let $f \in \mathcal { C } _ { b } ^ { 0 } ( [ 0 , + \infty [ )$ and $S \in \mathbb { R }$. Prove that $$\left( \lim _ { t \rightarrow 0 ^ { + } } \mathcal { L } ( f ) ( t ) = S \text { and } f ( t ) \underset { t \rightarrow + \infty } { = } O \left( \frac { 1 } { t } \right) \right) \Rightarrow \left( \int _ { 0 } ^ { + \infty } f ( x ) d x \text { converges and } \int _ { 0 } ^ { + \infty } f ( x ) d x = S \right)$$ For this, using the notations of question 5 of section 2, one can prove that there exist $M > 0$ and $A > 0$ such that for all $t > 0$ $$\begin{aligned}
\left| \int _ { A } ^ { + \infty } f ( x ) g \left( e ^ { - t x } \right) d x - \int _ { A } ^ { + \infty } f ( x ) P _ { 1 } \left( e ^ { - t x } \right) d x \right| & \leqslant M \int _ { A } ^ { + \infty } Q \left( e ^ { - t x } \right) e ^ { - t x } \frac { 1 - e ^ { - t x } } { x } d x \\
& \leqslant M \int _ { 0 } ^ { 1 } Q ( u ) d u
\end{aligned}$$
Let $f \in \mathcal { C } _ { b } ^ { 0 } ( [ 0 , + \infty [ )$ and $S \in \mathbb { R }$. Prove that
$$\left( \lim _ { t \rightarrow 0 ^ { + } } \mathcal { L } ( f ) ( t ) = S \text { and } f ( t ) \underset { t \rightarrow + \infty } { = } O \left( \frac { 1 } { t } \right) \right) \Rightarrow \left( \int _ { 0 } ^ { + \infty } f ( x ) d x \text { converges and } \int _ { 0 } ^ { + \infty } f ( x ) d x = S \right)$$
For this, using the notations of question 5 of section 2, one can prove that there exist $M > 0$ and $A > 0$ such that for all $t > 0$
$$\begin{aligned}
\left| \int _ { A } ^ { + \infty } f ( x ) g \left( e ^ { - t x } \right) d x - \int _ { A } ^ { + \infty } f ( x ) P _ { 1 } \left( e ^ { - t x } \right) d x \right| & \leqslant M \int _ { A } ^ { + \infty } Q \left( e ^ { - t x } \right) e ^ { - t x } \frac { 1 - e ^ { - t x } } { x } d x \\
& \leqslant M \int _ { 0 } ^ { 1 } Q ( u ) d u
\end{aligned}$$