Learn Danish: Hun gik uden at have sagt farvel. - She left without having said goodbye.

While at gå literally means "to walk" or "to go", it is very often used to mean "to leave" a place or gathering. For example, jeg går nu means "I am leaving now".

English uses the "-ing" word "having" after "without". Why does Danish use at have?

Danish does not have a direct equivalent to the English "-ing" form (the gerund) when following a preposition like "without". Instead, Danish uses the preposition followed by at and the infinitive of the verb. So, "without having" naturally becomes uden at have.

Could I just say uden at sige farvel instead of uden at have sagt farvel?

Yes, you can! Uden at sige farvel means "without saying goodbye" and is very common. The sentence here uses uden at have sagt farvel ("without having said goodbye") simply to emphasize that the action of saying goodbye was missing before she walked out the door.

Is it always at sige farvel for "saying goodbye", or do you use other verbs?

It is almost always at sige farvel. Just like in English, you use the verb for "to say" (at sige) when bidding someone farewell in Danish.