Let's consider two sequences $(a_n), (b_n)$ in $\mathbb{R}$ such that $$\lim_{n \to +\infty} a_n, \lim_{n \to +\infty} b_n = + \infty$$
Proposition: the sequence $$x_n = \left(1 + \frac{1}{a_n}\right)^{b_n}$$ has a limit if $\lim_{n \to +\infty} \frac{b_n}{a_n}$ exists.
How can one prove it? What is this limit? Does $x_n$ have a limit if $\lim_{n \to +\infty} \frac{b_n}{a_n}$ does not exist?