Actually I'm wrong about L'Hospital's Rule in this problem; it solves nothing here. I would insert an "[EDIT ...]" into my original post to avoid misleading people, but it's too late to edit.

It doesn't matter how many times you differentiate x^(1/3); you can not evaluate the result at zero. Which is funny, because you'd think you could differentiate x^3 and use the inverse function theorem. But there's that caveat about it not working when the derivative is zero.

I am now revisiting high school calculus and doubting my own sanity. Will post updates if/when I get my head right.