Limit of $\lim\limits_{n\to\infty} (1 + \frac{x_n}{n})^n$
This is actually a very useful fact it is proved as follows.
$$\left(1+\frac{x_n}{n}\right)^n= \left[ \left(1+\frac{1}{n/x_n}\right)^{n/x_n}\right]^{x_n}$$
And $$\left(1+\frac{1}{n/x_n}\right)^{n/x_n}\to e.$$
Whenever $a<x<b$, we have $\left(1+\frac an\right)^n<\left(1+\frac {x_n}n\right)^n<\left(1+\frac bn\right)^n$ for almost all $n$ and therefore $e^a\le e^x\le e^b$. The result then follows from contiunity of the exponential .