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

is

My book says that it comes down to showing that

So I have to find

I'm told that this is equivalent to finding

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.