That's correct. The limit of denominator and numerator are both infinity, so you have and can apply L'Hopital's rule. Taking the derivative of numerator and denominator gives and again you have . So apply L'Hopital's rule again to get . Since this goes to infinity as x goes to infinity, you can conclude that .
You probably didn't mean to say this in your first post, but I should point out that you can't assume that is infinity - it might be, but it could also be any positive number or zero.