A question about asymptotic notation
If $f(n)\sim g(n)$ then $\frac{f(n)}{g(n)}\rightarrow 1$ as $n\rightarrow \infty$ so as $n$ goes to $\infty$, $\frac{f(n)}{g(n)}=O(1) \implies f(n)=O(g(n))$
Exponential doesn't, take $f=x^2+x, g=x^2$ then $f\sim g$ but $$\frac{e^f}{e^g}=\frac{e^{x^2}e^x}{e^{x^2}}=e^x\rightarrow \infty$$