I'm stuck on the following problem from Rudin:

we're given: $\displaystyle f(x)=\int_x^{x+1} sin(t^2)dt$.

The first question was to prove that $\displaystyle |f(x)|<\frac{1}{x}$ if $\displaystyle x>0$. This was easy to prove if we let $\displaystyle u=t^2$ and then integrate by parts.

The second question is to show that $\displaystyle 2xf(x)=cos(x^2)-cos[(x+1)^2]+r(x)$ where $\displaystyle |r(x)|<\frac{c}{x}$ where c is a constant. Well, that was also easy to prove and we get c = 2.

Now, the third parts asks for the upper and lower limits of $\displaystyle xf(x)$ as $\displaystyle x\rightarrow\infty$. How can we do this?

Finally, we're asked whether $\displaystyle \int_0^{\infty}sin(t^2)dt$ converges or not. How can we know?

Thank you in advance.