Whats bigger? lim n->infinity n^x or lim n->infinity x^n

Question

What is bigger?

lim n->infinity n^x

or

lim n->infinity x^n

I have a relationship where I am trying to find the lim n->infinity (2^n + n^20) / 3^n and am having a hard time deciphering it.

Answer 1

What the question asks is not what you probably mean, since $$ \lim_{n\rightarrow\infty}n^{x}=\lim_{n\rightarrow\infty}x^{n}=\infty\text{ for }x>1. $$ What you probably care about is $$ \lim_{n\rightarrow\infty}\frac{n^{x}}{x^{n}}=0\text{ for }x>1. $$ In other words, $n^{x}$ is in $O(x^{n})$. See if you can prove this.

Answer 2

That later, $x^n$, is faster. You should get 0 as the limit after splitting the fraction and distributing the limit