I have this function
$$\frac{(5x^5+3x^2)(x\log x+x)}{10x^3}$$
and I have to give a big O estimate for it and so far here's what I have got.
I know to give a big O estimate, first I have to find the dominant term. In this case, after getting rid of slow increasing terms and simplifying, I am here:
$$\frac{x^3\log_2(x)}{2}+\frac{x^3}{2}$$
Then I can factor a $x^3$ out of the two fractions which I think is dominant term, so then the Big O estimate would be $O(x^3)$. Is this correct? I am mainly confused because of the $\log x$ in the function towards the end.
Thanks!
Best Answer
$O(x^3 \log x)$ will do it.
The function is not bounded by a constant multiple of $x^3$ because $\log x \to\infty$ as $x\to\infty.$