= Solution
By part (c) with $\lambda=1$, let $T_n=E_1+\cdots+E_n\sim\operatorname{Gamma}(n,1)$ for <independent and identically distributed> <exponential random variables>. The requested integral is $\mathbb P(T_n\leq n)$. Each summand has <expectation> one and <variance> one, so the <central limit theorem> gives
$$
\frac{T_n-n}{\sqrt n}\xrightarrow{d}N(0,1).
$$
Zero is a <continuity> point of the <standard normal distribution>, and consequently
$$
\boxed{\lim_{n\to\infty}\frac1{(n-1)!}\int_0^n x^{n-1}e^{-x}\,dx=\Phi(0)=\frac12.}
$$
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