How can I simplify $\lim_{n\to \infty } \, \int_0^n e^{-x} \left(1-\frac{x}{n}\right)^n \, dx$ ?
Limit[Integrate[(1 - x/n)^n/E^x,
{x, 0, n}], n -> Infinity]
The result was
Limit[ConditionalExpression[$\left.e^{-n} (-n)^{-n} (\Gamma (n+1,-n)-\Gamma (n+1)),\Re(n)>0\land \Im(n)=0\right],n\to \infty]$
Limit[ConditionalExpression[
(-Gamma[1 + n] + Gamma[1 + n,
-n])/(E^n*(-n)^n),
Re[n] > 0 && Im[n] == 0],
n -> Infinity]
was supposed to be $1$ since
$\int_0^{\infty } e^{-x} \, dx=1$
Integrate[E^(-x), {x, 0, Infinity}]
I did try Simplify and Reduce but neither work.
Thank you in advance
Assumptions
? $\endgroup$DiscreteLimit
? $\endgroup$