I'm trying to prove that $\lim\limits\_{m \to \infty}[(1 + \frac{1}{m})^m]^x = \exp(x)$ by expanding polynomials and comparing them, given that $\exp(x)$ is already defined as:

$$\exp(x) = \sum\_{n=0}^{\infty} \dfrac{x^n}{n!} = 1 + \dfrac{x}{1!} + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \dotsc$$

I tried going about this by firstly expanding the said limit with binomial expansion as
$$\lim\limits\_{m \to \infty}\left[\left(1 + \frac{1}{m}\right)^m\right]^x = \lim\limits\_{m \to \infty} \left(\binom{m}{0} + \binom{m}{1} \frac{1}{m} + \binom{m}{2} \frac{1}{m^2} + \dotsb \right)^x $$

my main goal being to obtain polynomial $P(x)$ which I can compare with $\exp(x)$. But here I get stuck because, after expressing each binomial coefficient, I'm not sure what to do about the $m \to \infty$ or later on exponent $x$ for that matter.

I'd really appreciate your help.