$\lim_{\epsilon\to0}\frac{\cos(\epsilon-n\frac{\pi}{2})}{\epsilon^n}$

Hint: $$\int_0^{\arccos{(0.5)}}\frac{\cos{(t)}}{t} dt\gt \int_0^{\arccos{(0.5)}}\frac{0.5}t dt\to\infty$$

Just for your curiosity.

Before starting, you can notice the problem close to $t=0$ since $$-\frac{\cos (t)}{t}=-\frac{1}{t}+\frac{t}{2}-\frac{t^3}{24}+O\left(t^5\right)$$ So, if you try to integrate between $0$ and $\epsilon$, you see the problem.

Sooner or later, you will learn that you face the problem of the cosine integral and that $$-\int_x^\infty \frac{\cos (t)}{t}\,dt=\text{Ci}(x)$$ Close to $x=0$, its series expansion is $$\text{Ci}(x)=\gamma+\log(x)+\sum_{k=1}^\infty \frac{\left(-x^2\right)^k}{2 k \,(2 k)!}$$