Denote by \(\lambda\) Lebesgue measure on \(\mathbb{R}\) and set

\[F(t):=\int_{(0, \infty)} e^{-x} \frac{t}{t^{2}+x^{2}} \lambda(d x), \quad t>0\]

Show that \(F(0+)=\lim _{t \downarrow 0} F(t)=\pi / 2\).

Remark. This exercise shows that 'naive' interchange of integration and limit may lead to wrong results: \(F(0+)\) is not zero!