I've just been looking at a proof of the product rule and there is one part which I can't get my head round:

Part of the proof states that when δx tends to 0, so does δu, δv and δy. I'm fine with this.

What I don't get is that when δx tends to 0, (δuδv)/δx also tends to 0. I don't see whats going on here (I never really learnt about limits formally) so if possible could somebody please try to explain why this is?

Thanks :]