Limit of an integral that arose in Fourier Analysis

Question

$$\lim_{\alpha\to\infty} \int_{0}^{\delta}\frac{\sin(\alpha x) \sin(\lambda\alpha x)}{x^2}dx$$ where $\lambda,\delta \in \mathbb{R}, \lambda,\delta > 0$. Appreciate your help in finding this limit. I stumbled upon this when i am trying to solve this which is a part of this bigger picture. Appreciate your general comments/suggestions on these as well.

Answer 1

Let $I(\alpha,\lambda)$ be the value of the integral. The change of variable $\alpha\,x=t$ gives $$ I(\alpha,\lambda)=\alpha\int_0^{\delta\alpha}\frac{\sin t\sin(\lambda\,t)}{t^2}\,dt. $$ Then $$ \lim_{\alpha\to\infty}\frac{I(\alpha,\lambda)}{\alpha}=\int_0^{\infty}\frac{\sin t\sin(\lambda\,t)}{t^2}\,dt=\frac{\pi}{2}\min(\lambda,1). $$