Big theta notation ($f=\Theta (g)$)

At the specified limit, big theta notation means that two positive constants give both-sided bounds:sufficiently near the limit. Equivalently, $f=O(g)$ and $g=O(f)$ in big O notation. In operation-count analysis, $3n^2+7n+4=\Theta (n^2)$ for $n\ge 1$, using $c=3$ and $C=14$. This supplies a tight scale without asserting that the normalized ratio tends to one.

For an example with no limiting ratio, let $f(n)=n(2+(-1{)}^n)$ for positive integers $n$. Then $n\le f(n)\le 3n$, so $f(n)=\Theta (n)$, but $f(n)/n$ alternates between one and three. A tight growth order is therefore weaker than asymptotic equivalence. The constant bounds also do not fix an exact factor by which a function changes when its argument doubles.

Donald E. Knuth's 1976 paper proposes using $\Theta$ for two-sided constant bounds in algorithm analysis. He credits Bob Tarjan and Mike Paterson with independently suggesting the notation. His discussion gives a concrete motivation for distinguishing an eventual upper estimate from a tight comparison. ^[ref]