# Intersection of functions given asymptotic relationship

If $f(n) = O(g(n))$ and $g(n) \neq O(f(n))$ than can we say that $f(n)$ and $g(n)$ will never intersect?

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• It's even possible for two functions to intersect infinitely often. (cc @YuvalFilmus) – Raphael Nov 12 '16 at 19:28

The asymptotic behavior of a function only depends on its value on "large" inputs. More formally, let $f$ be an arbitrary function, and let $f'$ be obtained by changing finitely many values of $f'$. Then for every function $g$, $f = O(g)$ iff $f' = O(g)$.