In my second-year calculus class this term, one of the thing that the professor insisted was wrong was that the limit of a two-dimensional function as the input approached a certain point could not be calculated simply by taking the limit of the function in every direction and verifying that they were all equal.
I've taken her word for it, but why is this not true? Is there a counterexample to this proposition, and if so, what general principle does it violate?