I faced with this equality $$\lim _{n \to \infty } \int_0^n \left( {1 - \frac{m}{n}}\right) ^n \log(m)dm= \int_0^\infty {{e^{ - m}}} \log mdm.$$ I know a rigorous proof for $\displaystyle \lim_{n\to\infty} \left(1+\frac {x}{n}\right)^n=\exp x$ but the problem is a general effect of limit on both integrand and upper bound and my question is proving the equality in the title and not just the example above; that is I can't rigorously prove why $$\lim _{n \to \infty } \int_0^n f(n,t) dt = \int_0^\infty \lim_{n \to \infty } f(n,t) dt.$$
Unfortunately, I don't know more than undergraduate real analysis. A simpler and clear proof would be much appreciated.