I made a statement in my coursework that the term with the largest growth rate for an arbitrary function f(x) is always the most simple g(x) in Big O/Theta/Omega Notation (removing coefficients). I believe this to be true, but I'd like more confirmation that my own assumptions.
Definition of Big Theta Notation:
f(x) = Θ(g(x)) as x → ∞ ⇔
∃M1 > 0 ∃M2 > 0 ∃x0 > 0 such that ∀x ≥ x0, M1 ⋅ g(x) ⩽ f(n) ⩽ M2 ⋅ g(x)
Suppose that $f(x) = n^2 + n + 1$. In this case, the highest growth rate would be $n^2$ and $g(x) = n^2$.
Suppose that $f(x) = 2n + 1$. In this case, the highest growth rate would be $n$ and $g(x) = n$.
Suppose that $f(x) = n^5 + n^4$. In this case, the highest growth rate would be $n^5$ and $g(x) = n^5$.
Is the statement "the term with the largest growth rate for an arbitrary function f(x) is always the most simple g(x) in Big O/Theta/Omega Notation (removing coefficients)" true?