I don't fully understand, why we use the big O to measure the time complexity of an algorithm. For example, suppose that the time complexity is $O(n!)$. Usually, we think that this is really bad. But wouldn't we write $ e^{-n} = O(n!) $ as well, which would mean that the time complexity decreases (even exponentially)!
That is, because for every $x \ge 0$, we have $$ e^{-n} \le n! , $$ so we write $e^{-n} = O(n!)$.