\inf \Big\{\lambda \in \mathbb{R} : \ e^{-\lambda x} \leq \frac{1}{1+x^2} \quad \forall x \in \mathbb{R}_+ \Big\}=\inf \Big\{\lambda \in \mathbb{R} : \ \lambda \geq \frac{\ln(1+x^2)}{x} \quad \forall x \in \mathbb{R}_+ \Big\}