Big O notation question of Kolman's book

Question

If

$$f(x) = x^{100} , g(x) = 2^x. $$

Show that $f(x)$ is a big $O(g(x))$, but $g(x)$ is not big $O(f(x))$.

Answer 1

Hint: find the limit

$$ \lim_{x\to \infty}\frac{x^{100}}{2^x} =? $$

and use the result

$$f=O(g)\quad \rm{iff} \quad \limsup_{x \to \infty}\frac{|f(x)|}{|g(x)|} =c,$$

where $c$ is finite.