I'd say it's hard to tell from that picture, but there are a few things to consider:-
The missing piece is from the free edge, it has not broken from the smile line, or where the tip joins the natural nail which are the most vulnerable points under stress. This would lead me to believe it was client inflicted, it's her right hand (I would guess dominant hand) and she's used her thumb to pick at something like removing a phone cover, or label from something.
However, from that picture it's hard to see if there is a sufficient stress point apex, you can't see how thin the product is at the tip, it should be thin but thick enough to create strength, for L&P to snap like that I suspect it was too thin.
I think it's a learning point for both of you, she needs to ensure she's not using them as tools and you need to ensure it's properly constructed.
You're right in that you cannot prevent clients misuse once they leave your salon but it depends on how you want your relationship with her to go. If she's a good client, discuss it with her and consider repairing it for free if you are in any doubt that it wasn't the best constructed nail, if when you see it you can categorically hand on heart say you did nothing wrong, then charge her.