First-Order Equations · Superchallenge problem

A mirror curve $y=f(x)$ ($x>0$) has this property: at every point $(x,y)$, the tangent line meets the $y$-axis at height equal to the distance from the origin to $(x,y)$. The curve passes through $(4,3)$. Find its limiting height $\displaystyle\lim_{x\to0^+}f(x)$.