Suppose I have a function $f(x)$ : $$f(x)=\int_{0}^{\infty}\beta(t)\sin(\frac{t\pi}{2})\cos(tx)dt.$$
Assuming $\beta(t)\sim\displaystyle\frac{1}{t^{3/2}}$ for $t\to+\infty$ is it true that for $x\to0$ $$f(x)\sim\int_{0}^{\infty}\frac{1}{t^{3/2}}\sin(\frac{t\pi}{2})\cos(tx)dt\,?$$ Namely, the values of the function $f(x)$ at a small value of $x$ are mainly dependent on the large $t$ behavior of their coefficient. How to prove it?
Thank you all in advance.