infinitesimal generator of reflecting Brownian motion

Question

Suppose $f\in C_0^{\infty}([0,\infty))$ and $f'(0)=0$. I'm having trouble proving that $$\frac{1}{t}E_x[f(|W_t|)-f(x)]\to\frac{1}{2}f''(x)$$ uniformly on $[0,\infty)$ as $t\downarrow0$. Showing the convergence at $0$ is easy enough using the Taylor expansion of $f$ about $0$. Additionally, I can show uniform convergence on $[\epsilon,\infty)$ for any $\epsilon>0$. This can be done by considering separately those paths that first hit $0$ before and after time $t$ and using the estimate $P_x(\tau_0<t)\leq2\exp(-\frac{x^2}{2t})$ while following the proof for non-reflecting Brownian motion.