calculus - Limit Proof of $e^x/x^n$

I am wondering how to prove

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

I was thinking of using L'Hospital's rule? But then not sure how to do the summation for doing L'Hospital's rule n times on the denominator? Or whether it would be easier using longs like $\lim_{x\to \infty} \ln(e^x)-\ln(x^n)$?

Thank you!

Answer

You can certainly use L'Hopital's $n$ times. That is, for each $n\geq 0$ we have

$$\lim_{x\to\infty}\frac{e^x}{x^n}=\lim_{x\to\infty}\frac{e^x}{nx^{n-1}}=\cdots=\lim_{x\to\infty}\frac{e^x}{n!}=\infty$$ since at each stage we are in $\frac{\infty}{\infty}$ indeterminate form.