How do LPs feel about fund managers investing their personal money? What kinds of limits are there? What do LPAs usually say?

A fund manager is a professional money manager and LPs expect them to primarily focus their energy on generating returns for them. LPs are generally not enthused if a fund manager's investing attention is focused otherwise.

LPAs are usually pretty restrictive with regards to fund managers investing personal money - it makes sense. Even if a fund has an LPA that is not as restrictive, a fund manager's biggest challenge (in personally investing into deals) is how they manage this with their LPs.
 
For example, if a personal investment has any overlap with fund's thesis, LPs are not likely to be happy. The fund manager will likely have to/should take any deal in this category to the fund's Investment Committee - and, irrespective of the result, it’s good practice to offer LPs the opportunity to invest (which will slow things down and may prevent the deal from getting done in the first place).

TL:DR: a fund manager should tread carefully and err on the side of the fiduciary responsibility owed to the fund.