I'm trying to compare $f(n) = 3^n$ and $g(n) = n\cdot2^n$ to determine whether $f \in O(g)$, $f \in \Omega(g)$, or $f \in \Theta(g)$.
My gut is telling me that $g(n) = n\cdot2^n$ grows faster, and so $f \in O(g)$, but I'm having a hard time coming up with a rigorous argument. I've tried using limits but keep getting things like $0 \cdot \infty$.
Can anyone lend a helping hand?
Best Answer
Hint: For intuition, try taking the limit of the ratio
$$\lim_{n\rightarrow\infty}\frac{n2^n}{3^n}=\lim_{n\rightarrow\infty}n\left(\frac{2}{3}\right)^{n}$$