Determine the highest order of contact at $x_0 = 0$
$f(x) = x^2$
$g(x) = \sin x$
My definition of contact is fuzzy. So I took the derivative at each step, and inspected when $f^{(k)}=g^{(k)}=0$
It's clear that $f^{(k)}(0)=0 \forall k$
$g'(0) = 1, g''(0)= 0, g'''(0)=-1, g''''(0)=0, ...$
Is the highest point of contact 2? or $\infty$?