For Julian III

Applying L'Hôpital's Rule $n$ times, we get $\begin{aligned}\displaystyle \lim_{x\rightarrow\infty} \frac{n! }{{e}^{x}}\\ \huge\color{#3D99F6}{=\boxed{0}}\end{aligned}$

I saw it as e^x grows faster than any other polynomial. It would approach 0 as the denominator grows faster than the numerator.

There are many ways to do this problem. I did it by repeated use of L'hospital.

Let $\displaystyle L =\lim _{ x\rightarrow \infty }{ \frac { { x }^{ n } }{ { e }^{ x } } }$

On differentiating the numerator and denominator n times, we get

$\displaystyle L =\lim _{ x\rightarrow \infty }{ \frac { n! }{ { { e }^{ x } } } } \\ \quad \quad =0$

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For MCQ lovers:

Since n is any constant, you can take $n=1$ . Then you'll have to differentiate it only once.

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Now the problem arises if n is very large. Then the limit would be infinity.

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