Is Intuition Enough to Determine Asymptotes in Calculus?

[monet A]

I am preparing to study for my exam, and there is something I am not sure about when evaluating the limit x --> Infinity of 4/x^2 - x
Intuitively the function seems to go to minus infinity, and I wonder if that is a sufficient answer or am I overlooking a rigorous method that should be applied here. It doesn't seem sufficient because I am looking for a horizontal asymptote and this answer doesn't guide me to a definitive point where one exists although it implies there would be one. Can someone give me a clue as to the method that would get me a clear answer.

Also as this function goes to 0, 4/0 is not defined so that's all I need to establish, right? And this would imply a vertical asymptote is at x = 0 which in turn implies that the function approaches infinity from above and below?

 

[HallsofIvy]

Yes, what you have said is correct. You might prefer to write this as a single
fraction: (4-x³)/x² which has a vertical asymptote at x= 0, goes to negative infinity as x goes to infinity, and goes to infinity as x goes to negative infinity.

 

[monet A]

So I can be certain at this point that there is horizontal asymptote.

I haven't looked at the graph I am trying to make certain that I can be confident in what I know from the equation without looking at the graph, as I would like to be in an exam.

I will be getting to looking at graphs to check my work next, believe it or not there is method to my madness.