From working on a problem I was lead to consider the function $\frac{T_n(n)}{e^n}$ where $T_n(x)$ is the $n$'th order Taylor polynomial of $e^x$.
Numerical evidence suggest that
$$\lim_{n\to \infty} \frac{T_n(n)}{e^n} \equiv\lim_{n\to \infty} \frac{\sum_{k=0}^n\frac{n^k}{k!}}{\sum_{k=0}^\infty\frac{n^k}{k!}} = \frac{1}{2}$$
Is there a nice proof for this statement? More generally: is there a 'standard' approach for evaluating limits on the form $\lim_{n\to\infty}\frac{f_n(x_n)}{f(x_n)}$ where $f_n$ is a series converging (uniformly) to $f$ and where $x_n$ is an unbounded sequence? I would also apprechiate refs. to similar questions on this site or in the literature (I could only find this one).