There's a very easy way to solve this. You can just take the lead coefficients, and see where they tend to go as you plug in smaller and smaller (more negative) numbers. That's going to be your limit because the other terms just get pretty much very small to matter at that point. So, for example, x² - 5x when plugged in -∞ would go towards positive infinity, because -5x just gets so small and irrelevant that it doesn't matter, and so you just care about what x² does. That's the easiest method I know of. Other ways include actually rationalizing the top and bottom but that's hideous.
