= Solution
The real-space expression $1/x$ means the <principal-value reciprocal distribution>. Apply even exponential damping. Oddness gives
$$
\widehat{e^{-a|x|}\operatorname{PV}(1/x)}(k)=-2i\int_0^\infty e^{-ax}\frac{\sin(kx)}x\,dx.
$$
Differentiate the integral with respect to $k$; its derivative is $\int_0^\infty e^{-ax}\cos(kx)dx=a/(a^2+k^2)$. Its value at $k=0$ is zero, so the integral equals $\arctan(k/a)$. Taking $a\downarrow0$ gives
$$
\boxed{\widehat{\operatorname{PV}(1/x)}(k)=-i\pi\operatorname{sgn}k.}
$$
The limit is a <tempered distribution> limit: the regularized transforms are uniformly bounded and converge away from zero, allowing dominated convergence against a <Schwartz function>. This <Fourier transform of a principal-value reciprocal> argument also establishes the needed Dirichlet integral $\int_0^\infty\sin u/u\,du=\pi/2$ in its Abel interpretation.
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