Express $4n +2\log^5 (n)$ in term of theta $\Theta$

Question

Express $4n +2\log^5 (n)$ in term of theta $\Theta$.

I had this question in a previous quiz, and I am still not able to figure it out. The professor tried to explain it to me, but I am still not sure how to go about it. I know the answer would be $\Theta (n)$ after taking the limit $n\to\infty$ of $ 4n / 2\log^5(n) $ which happens to be $0$. Can someone walk me through the whole thing, step-by-step?

Thanks in advance.

Answer 1

You are not quite right.

What you should do is:

$$4n+2\log^5(n)=n\left(4+\frac{2\log^5(n)}n\right)$$

and because

$$\lim_{n\to\infty} \frac{2\log^5(n)}n=0,$$ you get that

$$4+\frac{2\log^5(n)}n$$

is bounded, so:

$$4n+2\log^5(n)=\Theta(n).$$