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On my setup it appears as though the limited preview rectangle, which shows as a pop-up when the mouse hovers above a question and contains a text excerpt of the question's main text, doesn't support TeX. Is it supposed to work like that or is this a bug?

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Yes, it's supposed to work like that. The hover effect is achieved by giving the <a> (link) tag a title property containing the excerpt, and then the tooltip is produced by the browser. There's no way (on SE's end anyway, the browser could theoretically be changed to do it) for this tooltip to support MathJax: the browser just reads the title attribute as plain text.

I guess it could be possible to modify everything to produce a tooltip with some Javascript and have this new tooltip be made to support MathJax, but frankly that sounds rather complicated for a small gain – you can just read the MathJax code to understand what's going on, and if you want to read more you can just click on the question. It would also cause the same problems as the search, where formulas are cut off in the middle and not rendered properly.

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    $\begingroup$ In principle it could be changed on SE's end by replacing standard title attributes by custom tooltips, like the one they use for Impact number in the profile and a few other fields. Those are regular divs that are made to appear on hover, and could be mathjax-processed as the rest. Of course, one probably would not want to have randomly cut-off formulas rendered in weird ways, anyway. $\endgroup$ – user147263 Mar 21 '16 at 15:54
  • $\begingroup$ @404 Well yes, maybe something like the tags. But honestly that sounds like a lot of complexity for a very small gain. Also this way even prehistoric browsers (and search engine crawlers, I guess) can read the title attribute. $\endgroup$ – Najib Idrissi Mar 21 '16 at 15:59

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