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
$$
\boxed{
\lim_{\lambda\downarrow0}
\bigl[f_+(\lambda)-f_-(-\lambda)\bigr]
=-2i\frac\pi2=-i\pi}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 327 / 1 / b / v / Solution

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A one-sided limit restricts the argument to values above or below the limiting point. It is denoted $\lim_{x\downarrow a}f(x)$ from above and $\lim_{x\uparrow a}f(x)$ from below.


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