Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2008 / Paper 1 / 19 / a / iii

For $n=1,2,3$, the difference quotient at zero is $x^{n-1}\sin (1/x)$. It oscillates without a limit for $n=1$, while the squeeze theorem gives limit zero for $n=2,3$. The other two cases fail continuity and therefore cannot be differentiable. Hence