Solution
= Solution
For $\lambda>0$, subtracting the two formulas from part (iv) yields
$$
f_+(\lambda)-f_-(-\lambda)
=\int_\lambda^\infty
\frac{e^{-ix}-e^{ix}}x\,dx
=-2i\int_\lambda^\infty\frac{\sin x}{x}\,dx.
$$
Using the stated <Dirichlet integral> and taking the <one-sided limit> gives
$$
\boxed{
\lim_{\lambda\downarrow0}
\bigl[f_+(\lambda)-f_-(-\lambda)\bigr]
=-2i\frac\pi2=-i\pi}.
$$