I've been working on this limit for a long time and just don't know how to show this. I tried representing h and d as vectors, but I cannot seem to accurately describe them. What would be a good method to show this?
HINT: Suppose we write $C$ as a graph over the tangent line at $p$, so we represent $C$ as $y=f(x)$, with $f(0)=0$ and $f'(0)=0$. Do you know (or can you derive) a formula for $\kappa$ in terms of derivatives of $f$? Can you express $2h/d^2$ in terms of $f(x)$ and find the limit as $x\to 0$?