C2 - Behaviour of polynomials

Q: For this curve, find where the curve meets the axes, state any symmetry and the behaviour as x -> infinity. Find the coordinates of any stationary points, determine whether they are maximum or minimum points or points of inflexion, and then sketch the curve.

The curve is y=3x^2-x^3

I have done everything above apart from the behaviour part, as I do not understand it. I would really appreciate it if someone explains to me how to do it.

Look at the graph of your function or simply look at the equation, what happens when x gets really really big?

It gets closer to infinity?....

Does it really? Does it get closer to infinity or -infinity?

Oh, it gets closer to -infinity, my mistake. But why is this? I think I know why but I can't really put it to words.

Oh, it gets closer to -infinity, my mistake. But why is this? I think I know why but I can't really put it to words.

Note that $y=3x^2-x^3=x^3(\frac{3}{x}-1)$.

Now as $x \to \infty$, we have that $x^3 \to \infty$ but what happens to the term in the bracket?

It gets closer to infinity?

Sorry, I have never done this topic.

It gets closer to infinity?

Sorry, I have never done this topic.

So when $x$ goes off to infinity, what atsruser has written is $\infty (0 -1) = \infty (-1) = -\infty$, essentially.

Not really. What I've written is $\text{(a very big number)} \times -1 = -\text{(a very big number)}$ and that's true regardless of how big the very big number happens to be.

Nah, you have $\frac{3}{x} \approx 0$ for really big $x$, you agree with me? So when $x$ goes off to infinity, what atsruser has written is $\infty (0 -1) = \infty (-1) = -\infty$, essentially. It's not really a topic, it's more of a logical thing you need to understand and be able to deduce yourself.

Thanks. I understand it slightly more now. So, the answer is just -infninity?

Kind of, you're meant to say that y diverges off to -infinity, yeah.

I checked the answer to this question and it was written minus/plus infinity. Is this the same thing as - infinity?

I meant "when you plug $x = \infty$ into what atsruser wrote"

They're going to beat you with a stick at Cambridge, if you write stuff like that

I half considered adding a spoiler to disclaim that I know that I'm being completely unrigorous.