Prove $\lim_{\Delta t \to 0} \frac{2}{\Delta t} \left(1-\Phi\left(\frac{\epsilon}{\sqrt{\Delta t}}\right)\right) = 0$

Question

Given $$\lim_{\Delta t \to 0} \frac{2}{\Delta t} \left(1-\Phi\left(\frac{\epsilon}{\sqrt{\Delta t}}\right)\right) $$

with $\Phi$ standard normal CDF, how can I prove the limit to be equal to $0$?

My attempt focused on the L'Hospital's rule but I can't exactly get the result. I also thought that since $\Phi$ is a CDF, its values are bounded between $0$ and 1, and maybe I could use this information somehow. Any hint?

Answer 1

Hint: $$\Phi(x) = \frac1{\sqrt{2\pi}}\int_{-\infty}^x x^{-s^2/2}\,ds$$ $$\Phi'(x) = ...$$ (use the Fundamental theorem of calculus