Simplifying $f(x) = \left(x^{3} + 2x^{2} + O(x)\right)\cdot\left(1 + \frac{1}{x} + O\left(\frac{1}{x^{2}}\right)\right) $

Question

Simplify $$f(x) = \Big(x^{3} + 2x^{2} + O(x)\Big)⋅\Bigg(1 + \frac{1}{x} + O\bigg(\frac{1}{x^{2}}\bigg)\Bigg) $$ as $x \to +\infty$.

I am a bit stuck as to what to do with the three sets of Big-Os I have at the end. Am I meant to just keep the lowest one?

I end up with these values

$$x^3+3x^2+2x+O(x)+O(1)+O\left(\frac{1}{x}\right)$$

Your help is appreciated.

Answer 1

Remember that $O(x^p)\subset O(x^q)$ whenever $p\leq q$ (assuming Big-O is understood as $x\rightarrow\infty$).

In other words, statements like $f(x)=O(x)+O(x^2)$ are redundant; we can just write $f(x)=O(x^2)$.