To prove a relation for a smooth, asymptotic plane curve, in arc length parametrization.

Question

Given a smooth plane curve, parametrized in arc length as $\alpha(s) \equiv (x(s),y(s))$ and given that $$\lim_{s \to \infty} \frac{y(s)}{s} = k,$$ $k$ a constant, and $$\lim_{s \to \infty}x(s) = 0,$$ I'd like some help to prove that (if possible)

$$\lim_{s \to \infty} \frac{dx}{dy} = 0.$$

Answer 1

Counterexample (not parametrized by arc length): $$ \alpha(t)=\Bigl(\frac{\sin(t^2)}{t},t\Bigr). $$