To simplify the question, we start with standard Brownian motion(BM) $B_t$. Then, $$\lim_{\epsilon \to 0} \frac{1}{\epsilon}\mathbb P( 1- \epsilon <B_1 < 1 ) = \phi(1),$$ where $\phi$ is the density function of the standard normal distribution.

Next, let $B^1_t$ be the killed BM at $1$, i.e. $$B^1_t = B_{t\wedge \tau}, \hbox{ for } \tau = \inf\{t>0: B_t \ge 1\}.$$ [Q1] Does the following identity hold: $$\lim_{\epsilon \to 0} \frac{1}{\epsilon}\mathbb P( 1- \epsilon <B^1_1 < 1 ) = 0?$$ [Q2] If [Q1] is yes, can we replace $B_t$ by any other diffusion process starting from zero?

Remark: This question is related to the reason why the fokker-planck equation of a killed diffusion imposes zero boundary condition. See Fokker-Planck equation for a truncated process