Learn Danish: Katten skubbede koppen ud til kanten og var lige ved at vælte den. - The cat pushed the cup out to the edge and was about to knock it over.

The phrase være lige ved at followed by an infinitive is a common way to express that an action is imminent or on the verge of happening. Because this sentence is in the past tense with var, it means the cat came dangerously close to knocking the cup over, but had not actually done it yet.

Why do we use ud til kanten instead of just saying til kanten?

Adding ud (out) emphasizes the direction of the movement. While til kanten simply means 'to the edge', ud til kanten gives a clearer visual of the cup being pushed outward from the center of the surface toward its boundary.

Does the verb vælte mean 'to fall over' or 'to knock over'?

It can mean both! In Danish, vælte is used both when something tips over by itself and when someone actively knocks it over. Here, because it is followed by the object den (it), we know it means the cat is actively knocking it over.

Why does the sentence end with the pronoun den instead of det?

The word den means 'it' and refers back to koppen (the cup). Since en kop is an en-word (common gender), you must use den to refer to it. You would only use det if the pronoun referred back to an et-word like glasset (the glass).