I can't reverse!

[Snare123]

I have a manual 2014 A3 8V and recently I tried to reverse but it wouldn't engage with the reverse gear. The reversing light comes on and I hear the usual "beep" but I can let the clutch out entirely without the gear ever engaging. 

I did struggle to get into reverse occasionally before this but nothing severe, maybe 1 in every 20 tries it didn't engage fully and I got a brief grinding noise (before immediately backing out of the gear and going back in sucessfully.) 

The car drives forward with no issues at all, from my very limited knowledge and some googling it seems like it could be something to do with the transmission but I'm far from certain and couldn't find anybody describing the exact issue I'm having. 

Any advice or insight would be appreciated, the car is under warranty for the next month but max claim is £1000, I know anything gear related can be expensive so can anyone advise what the issue might be and how much I should expect to pay to fix it. 

My other question is whether or not its worth taking it to an Audi garage or will I be charged an extortionate amount for any diagnostics/repairs if I do? The warranty covers repairs at any garage and they negotiate with them directly on price but with the limit almost certainly being hit I want to keep costs down if possible. 

Thanks