Problem: Consider the following integral:$I(x)=\int_0^\infty \frac{t^{x-1}e^{-t}}{t+x}dt$In the limit$x\to\infty$, find approximate behavior of the integral up to and including the leading order in x. Provide your answer in a $\boxed{}$ latex environment.
Solution: $\boxed{I(x)\sim \frac{\Gamma(x)}{2x}}$