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This document (page $5$ and $6$, it acutally starts from example $3$) shows that for a constant potential semi-infinite plane given by $\,y=0\,$ and $\,x>0~$the $E$-field may be written as:

$$\textbf{E}=\frac{\sqrt{2}}{4\sqrt{x^2+y^2}}\Bigg (\sqrt{\sqrt{x^2+y^2}-x},\frac{y}{ \sqrt{\sqrt{x^2+y^2}-x}}\Bigg )$$

Then the author went on to say the $E$-field just above the sheet and just to the right of its edge is $$\textbf{E}(\epsilon , \delta)\approx\Big (0, \frac{1}{2\sqrt{\epsilon}}\Big )$$

This result seems to condratic the analytical expression of $\,\textbf{E}\,$ above, for example, if we plug $\epsilon$ and $\delta$ in the first component, we 're essentially going to find the same expression but with $\epsilon$ and $\delta$ instead of $x$ and $y$.

Is it just an oversight from them or am I missing something?

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It looks to me like this is a case where you can't freely interchange the order of the limits, and the claim holds only when $y \ll x$. Let $r=\sqrt(x^2+y^2)$ and $\alpha=r-x$. Then for $y \ll x$, $\sqrt{\alpha}\sim yx^{-1/2}$, which is small.

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The author has taken the limit as $\delta\to 0+$, while keeping $\epsilon$ merely small.

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