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My book "Electromagnetic Fields" says in $\text{Section}\ 3.4$:

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Question

Why does the limit in equation $(3.42)$ exist (convergent)? Why is the contribution from $(S-S_{\delta})$ remains bounded?

Please provide a proof.

My try

I tried to use polar coordinates but still $|\mathbf{r}-\mathbf{r}'|^{-1}$ type singularity remains.

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Note:

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