Stirling's formula Baby Rudin

I have some questions I would be grateful for any kind of help.

$(1)$ why do we define h function and then why $h(0)$ should be equal of 1.

$(2)$ why do we have integral from -$\infty$ to $\infty$ in $(107)$.

(1) Rudin wants a function $h$ that satisfies $(1+u)e^{-u} = \exp\left(-\frac{u^2}{2}h(u)\right)$. The expression Rudin got for $h$ is $h(u) = \frac{2}{u^2}(u-\log(1+u))$ which is not defined for $u=0$, but $\lim_{u\to 0} h(u) $ exists, so the the function can be extended to be continuous at $u=0$ defining it at $u=0$ as the value of such limit, which is $1$.

(2) After the substitution $u=\sqrt{2/x}$ you get the integral $\displaystyle\int_{-\sqrt{x/2}}^\infty \psi_x(s)\;ds$, but $\displaystyle\int_{-\sqrt{x/2}}^\infty \psi_x(s)\;ds = \int_{-\infty}^\infty \psi_x(s)\;ds$ because $\psi_x(s)$ is defined as $0$ for $s\le -\sqrt{x/2}$