It's well-known that $\lim_{x\rightarrow\infty}\left[1+\frac{a}{x}\right]^x=\operatorname{exp}[a]$. I am wondering how fast does the limit converge as $x$ increases, and how the speed of convergence depends on $a$. That is, I would like to find out what $f(x,a)$ is where:
$\left|\left[1+\frac{a}{x}\right]^x-\operatorname{exp}[a]\right|=\mathcal{O}(f(x,a))$
However, I'm having trouble evaluating that absolute value. Any tips? Perhaps this a known result...