there are a lot of ways to show this. sometimes you can simplify so that you can plug in, sometimes not. give us the particular example so we can help you

note: you are correct, you cannot prove that the limit exists by showing that it is the same along finitely many lines. however, to show that it doesn't exist, you have to show that you get a different answer along some path. if you keep getting the same thing, it suggests that the limit exists and that it is that number, but you can't tell for sure, you have to prove it