I read that in order to calculate the Fresnel integrals $\int_{0}^{+\infty} \cos(t^{2}) dt$ and $\int_{0}^{+\infty} \sin(t^{2}) dt$, we could set $A(x) = (\int_{0}^{x} \cos(t^{2}) dt) \times (\int_{0}^{x} \sin(t^{2}) dt)$ and $B(x) = (\int_{0}^{x} \cos(t^{2}) dt)^{2} - (\int_{0}^{x} \sin(t^{2}) dt)^{2}$, and then find the following limits:
$\lim\limits_{\lambda \to +\infty} \frac{1}{\lambda} \int_{0}^{\lambda} A(x) dx$ and $\lim\limits_{\lambda \to +\infty} \frac{1}{\lambda} \int_{0}^{\lambda} B(x) dx$.
But so far, I was not able to calculate these limits ...
Thank you for your help !