exponential time complexity

Question

given that:

$f(n) = a^n$ and $g(n) = b^n$

where, a,b are positive integers and n is a positive real number

does there exist some $f(n) \notin \mathcal{O}g(n)$?

ie. $f(n) \leq c_1 \cdot g(n)$ is false?

Answer 1

As long as $a > b$, $f(n) \not \in \mathcal{O}(g(n))$.

Answer 2

Let $a=4$ and $b=2$. We show that $a^n \not\in O(b^n)$. So we need to show that for every constant $c$, there are infinitely many $n$ such that $4^n \gt c(2^n)$. Equivalently, since $4^n=(2^n)(2^n)$, we want to show that there are infinitely many $n$ such that $2^n \gt c$. This is clear, since $2^n\to\infty$ as $n\to\infty$.