Show that $\lim_{n\to \infty}\int_{[0,\infty]}\left(1+\dfrac{x}{n}\right)^n\exp(-ax)=\dfrac{1}{a-1}$

Note that

$$\left(1+\frac xn\right)^n\le e^x$$

for $x>0$ and $n\ge 0$. Therefore, the Dominated Convergence Theorem guarantees that we have

$$\lim_{n\to \infty}\int_0^\infty \left(1+\frac xn\right)^n\,e^{-ax}\,dx=\int_0^\infty \,e^{-(a-1)x}\,dx=\frac{1}{a-1}$$

for $a>1$.