‘at the limit’ is an adverb used to convey that something is ‘at the most (A)’, or ‘at best (A)’. This just indicates that the point that (A) has reached cannot be progressed beyond, due to having already reached some physical or metaphorical barrier. This can also often come across as ‘no more than (A)’, especially in cases where no actual barrier exists, but the speaker cannot imagine that anything higher (or lower) than (A) is possible.
is an adverb, it often appears at the beginning of sentence, or at the beginning of the second clause of a sentence, before the thing that is thought to be at its maximum is described.
This bag probably costs 1000 yen at most.
You need to work for 3 months at the most.
This is at best what I can cook.
Since I am working now, even if I could study, I can only study 1 hour a day at most.
Fun-fact - 精々（せいぜい） is said to have come from the old-fahioned word 精試（せいぜい）, which meant ‘to completely exhaust all of one’s good faith’. In modern times, 精々（せいぜい） itself has kept this meaning of ‘completely exhausting (A)’. However, this is no longer used in reference to ‘good faith’, but rather exhausting all of one’s ‘power’, ‘mental capacity’, or ‘power of assumption’ in relation to (A).