I have the following limit $\displaystyle\lim_{n\to\infty}\frac{e^n}{n!}$. Now if I try to solve this using this using L'hopital's rule, I won't be able to since I can't take the derivative of $n!$.
My question is why can't I take the derivative? Generally speaking, why can't I take the direct derivative of a factorial? I've seen other questions, but I just want a more simple answer (for someone at the Calculus 1 or 2 level).
Thank you in advance
Best Answer
A derivative of the factorial function exists if you can define factorials of non-integers is a smooth way, and that can be done by using the fact that $n!=\int_0^\infty x^n e^{-x}\,dx$. But actually writing down a good expression for the derivative is another matter.
However, the limit is easy to show to be $0$. Think of what happens when $n$, on its way up to $\infty$, goes from $1000$ to $1001$, and observe that the pattern continues: The numerator gets multiplied by $e$, making it less than $3$ times as big, but the denominator gets multiplied by $1001$, so the whole thing gets multiplied by something smaller than $3/1001$. And at the next step, from $1001$ to $1002$, and all later steps, it gets multiplied by something even smaller. And this keeps happening over and over every time $n$ increases by $1$.
So the fraction must approach $0$.