I'm going to assume (I think a locally bounded derivative would be enough).

Obviously your limit is when . Since (I'm assuming this is your definition of simple zero) exists, so if we put we get

and the first term goes to while the second blows up so the limit doesn't exist. If in the above, calculting we arrive at the conclusion that the limit in this case is .

Now if we get, using the triangle inequality first, the fact that is locally Lipschitz continous and that

So in fact is the only value for which the limit isn't zero or infinite.