Is it possible to prove from the definition of Big O that $2x^3+x^2logx$ is $O(x^3)$? I''m stuck at it please explain me this one.

Question

Is it possible to prove from the definition of big $O$ that $2x^3+x^2logx$ is $O(x^3)$?

Answer 1

You have that $\log x \leq x$ and hence $$2x^3 + x^2 \log x \leq 2x^3 + x^2\cdot x = 2x^3 + x^3 = 3x^3$$ and therefore $2x^3 + x^2\log x$ is $\mathcal{O}(x^3).$