How can I find the fundamental solution of the wave equation in 3D?

Question

After taking Fourier transforms of $$ u_{tt} - \Delta u = 0 $$ and computing the inverses, I arrive at the following formula for the fundamental solution in $n$ dimensions in the sense of distributions: \begin{align*} R(t, x) &= c'_n \Im \lim_{\epsilon \to 0}(|x| ^2 - (t - i \epsilon) ^2) ^{-(n-1)/2}, \end{align*} where the $c'_n$ are some constants. I have checked this against Taylor's PDE book 1, Chapter 3 and it is correct. Now, when $n = 3$, this formula reduces to, using $\Im z = (2i)^{-1}(z - \bar z)$: \begin{align*} R(t, x) &= \frac{1}{4 \pi ^2 i} \lim_{\epsilon \to 0} \left((|x| ^2 - (t - i \epsilon) ^2) ^{-1} - (|x| ^2 - (t + i \epsilon) ^2) ^{-1} \right) \\ &= \frac{1}{4 \pi ^2 i} \lim_{\epsilon \to 0} \left((|x| ^2 - t ^2 + 2 i \epsilon t + \epsilon ^2) ^{-1} - (|x| ^2 - t ^2 - 2 i \epsilon t + \epsilon ^2) ^{-1} \right) \\ &= - \frac{1}{2 \pi} \delta(|x| ^2 - t ^2), \end{align*} where I used the Plemelj jump function in the second line. From here I have two problems, first, how can I justify using the Plemelj jump function given that I have the $\epsilon^2$ term vanishing in the real part? Is there an argument that allows me to avoid having to prove the identity again in the case where the point on the real under the jump is converging to some other point? Second, how do I get from my formula to the one given in the book which is $$ R(t, x) = (4 \pi t)^{-1} \delta(|x| - |t|) \ ? $$ Actually I just realized I don't even know what $\delta(|x|^2 - t^2)$ means. Formally, $\delta$ is just a linear functional on Schwartz functions, so how can I even compose it with another function?