One of the starkest differences between dealing with differentiation of a multi-variable function and a single variable function is that there are now an infinite number of ways to take a limit, and it becomes much more difficult to show that the derivative exists at a point by checking that derivatives exist along all possible paths. With functions in two variables, for example, one can usually play around with and separately to try to get different values at a limit point. Recently I ran across a problem that I found striking.
Let be positive real numbers and let
Prove that if and that the limit does not exist if
I found this particularly interesting because it is not at all clear why the quantity should control the existence of the limit at . The proof, I thought, was quite clever.
In retrospect the case is easier, as we just have to exhibit a path for which the limit as approaches along the path is non-zero. Indeed, trivially we have for any positive values of , and so this suffices. Now we choose the path where . Then we have
and this limit is plainly if and if .
The other situation is much trickier. We will require two ad-hoc inequalities.
Now we have
and if then the right hand side tends to as , thus establishing our proposition.
along this path.