<s>[INST] n! v.s. $a^{n}$ How can we know which one is faster without graphing? n! v.s. $a^{n}$ If we are given an arbitrary number a (a>1). How can we know which one is faster as n->INFINITY without graphing? 
 [/INST] For $n \rightarrow \infty$, $n!$ is always going to win. We have
\begin{align*}
\frac{a^n}{n!} & = \frac{a}{1} \frac{a}{2} \ldots \frac{a}{n-1} \frac{a}{n}
\end{align*}
Now, let $N$ be the smallest integer that satisfies $N \geq a$. Furthermore, let $n > N$. We have
\begin{align*}
\frac{a^n}{n!} & = \underbrace{\frac{a}{1} \frac{a}{2} \ldots \frac{a}{N-1}}_{>1} \times\underbrace{\frac{a}{N} \frac{a}{N+1} \ldots \frac{a}{n-1} \frac{a}{n}}_{<1}
\end{align*}
where the tail becomes smaller and smaller for increasing $n$. You can now verify that the tail of this product 'overwhelms' the head for $n$ sufficiently large.
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