How do I prove that $e^x < x^4$ as $x\to \infty$?

Question

Basically, how do I prove $x^4>e^x$ as $ x \to \infty $?

Answer 1

This is wrong.

Correct inequality is $\color{blue}{x^4 <e^x}$

You can see the limit as $x \to \infty$

$$\lim_{x \to \infty} \frac{x^4}{e^x}=0$$

(By 4 times applying L'Hopital's rule)

Answer 2

If you expand $e^x$ for positive $x$ and examine the term $\cfrac {x^{r+1}}{(r+1)!}$ which is one of an infinite number of positive terms, you can conclude that $$\frac {e^x}{x^r}\gt \frac x{(r+1)!}$$

So $\cfrac {e^x}{x^r}$ is unbounded for all positive integers $r$ as $x\to \infty$