I am attempting to tackle a problem in a computer science course which involves taking a limit. I had gotten a piece of advice but I'm not quit sure about some of the details and I was hoping someone could give me some clarification.

The problem is...

Prove the is $\displaystyle n\sqrt{\log{n}}$ is $\displaystyle \omega(\sqrt{n}\log{n})$

My book says that it comes down to showing that $\displaystyle \lim_{x \to \infty} \frac{f(n)}{g(n)}} = \infty$

So I have to find $\displaystyle \lim_{x \to \infty} \sqrt{\frac{n}{\log{n}}}}}$

I'm told that this is equivalent to finding$\displaystyle \lim_{x \to \infty} \frac{n}{\log{n}}}}$ since the square root is monotonic.

I can use L'Hopital's rule and show that the limit converges to infinity. I do understand why the square root function is monotonic but I don't understand why I'm allowed to disregard the affect of the square root on the function.

Could someone explain why?