I wonder if this doesn't make sense.If it is an unspecific library, you can use "a library" and for a specific library, "the library". My acquaintance answered with this, but got no mark for this part in a midterm of a Korean middle school. I need your clear answer.
ex)I wonder whether I can borrow the book at a library.
It's entirely understandable. Personally, I would say "from the library", but you are "at a library" at the point where you take the book to the desk and actually go through the borrowing procedure.
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.
But the question 'I wonder whether I can borrow the book at a library' means 'I wonder whether there's a library somewhere where I can borrow the book' - perhaps for some reason the speaker thinks it's not the sort of book a library would stock.