How can I calculate the limit of exponential divided by factorial? I suspect this limit is 0, but how can I prove it?
$$\lim_{n \to +\infty} \frac{2^{n}}{n!}$$

To prove $\lim_{n\to \infty} \,\frac{n!}{a^n} = +\infty$ it is suffice to prove that for all $k>0$ it is possible $\frac{n!}{a^n} > K$ for almost $n$, that is $n!>K a^n$. This means
$1\cdot 2\cdot\ldots\cdot\frac{n}{2}\cdot\ldots \cdot n > K \cdot a\cdot \stackrel{\stackrel{n}{\smile}}{\ldots}\cdot a,$
that is
$\frac{n/2+1}{a}\cdot\ldots\cdot \frac{n}{a} > K\cdot  \frac{a}{1}\cdot\frac{a}{2}\cdot \stackrel{\stackrel{n/2}{\smile}}{\ldots}\cdot \frac{a}{n/2}.
$
Note that
$\frac{n/2+1}{a}\cdot\ldots\cdot \frac{n}{a} >
\left(\frac{n/2+1}{a}\right)^{n/2} > K \cdot a^{n/2} >
 K\cdot  \frac{a}{1}\cdot\frac{a}{2}\cdot \stackrel{\stackrel{n/2}{\smile}}{\ldots}\cdot \frac{a}{n/2}$
which is true for $n$ verifying $\frac{n/2+1}{a} > 2 a,$ because $K^{2/n}=\sqrt[n]{K^2}\to 1$ and $2a>K^{2/n}a$ for $n$ large enough.
