limits - Is the derivative of a exponential function a^x always greater than the derivative of a polynomial x^n as x approaches infinity

with n and a being any constants > than 1.

I have tried taking the $\lim\limits_{x \to \infty} a^x / x^n$, and l'hopitals is telling me than $x^n$ can always be reduced to 1 with multiple iterations, so the limit is always infinity, and $a^x$ always grows faster than $x^n$

Answer

Your argument using L'Hopilat rule is correct you need just to add the condition $a>1$

For $a> 1$ it's true that $a^x$ is very larger then $x^n$ to see this you compose with a logarithm:

$$\lim_{x\to \infty} \frac{a^x}{x^n}=\lim_{x\to \infty} e^{\displaystyle x\ln(a)-n\ln(x)} =e^{+\infty}=+\infty$$

because the linear functions are always larger than logarithmic functions.