Show that the Gamma integral exists for $x > 0$.

You are integrating a function that is continuous on $(0,\infty)$. You just have to check the integrability at the endpoint of the interval.

• At $0$ you have $u^{x-1}e^{-u} \underset{u \rightarrow 0+}{\sim} u^{x-1}$ which is a known integral. It converges if and only if $1-x < 1$.

• At $+ \infty$ you don't have an equivalent that would be that simple but still the exponential dominates everything. More rigourously, you know that $$ u^2 u^{x-1}e^{-u} = u^{x+1}e^{-u} \underset{u \rightarrow \infty}{\longrightarrow} 0 $$ so there exists $A > 0$ such that forall $u > A$, $u^{x-1}e^{-u} \leq \frac{A}{u^2}$. The right hand side is integrable at $+\infty$ and everything is non negative so your integral converges at $+\infty$.