Let $f\left(t\right):={\int }_0^{\infty }{x}^{-2}{\sin }^{2}x{e}^{-tx}dx,t⩾0$.

(i) Show that $f$ is continuous on $[0,\infty )$ and twice differentiable on $(0,\infty )$.

(iii) Use (i) and (ii) to obtain a simple expression for $f$.

(ii) Find f" and work out the limits lim, f (t) and lim, f' (t).