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?
From the definition of real power $$x_n = \left(1 + \frac{1}{a_n}\right)^{b_n} = \exp\left(b_n \ln \left(1 + \frac{1}{a_n}\right)\right) = \exp\left(\frac {b_n}{a_n} a_n \ln \left(1 + \frac{1}{a_n}\right)\right) = \exp\left(\frac {b_n}{a_n} \ln \left(1 + \frac{1}{a_n}\right)^{a_n}\right)$$ But the exponential function is continuous, so $$\lim e^{p_n} = e^r \iff \lim p_n = r$$ Notice that $$\lim \frac {b_n}{a_n} \ln \left(1 + \frac{1}{a_n}\right)^{a_n} = \lim \frac {b_n}{a_n}$$ which gives us the thesis