I know a similar problem in demidovich's problem set #2357 about proving $$\lim_{x \to 0^+}x^a\int_{x}^1 \frac{f(t)}{t^{a+1}}dt$$it proves by dividing the integral into two parts and used two inequality to prove it less than some $\epsilon$. But the method does not apply.
My other attempts discovered $\sin(x)$ is not arbitrary and cannot be casually changed to some constant even on some intervals.
After Taylor expansion of $sin(x)$ is applied, the limit is still hard to calculate since it is a sum of an infinite series of improper integrals.