Does the integral $$ \int_0^\infty 1 -\exp\left(-\frac{1}{2x}\right) dx $$ exist?
I see that there is no problem for $x = 0$ as there is a continuous extension. So the question remains whether $$ \lim_{T \to \infty} \int_T^\infty 1 - \exp\left(- \frac{1}{2x} \right) dx = 0. $$