I've cut roots when digging palms before, if you only broke one I'm sure the palm will be fine. Mexican fan palms tend to be rugged palms (especially their seedlings!), it might take a while for the palm to recover from being dug but I'm sure that it will eventually recover. Just water it thoroughly and in a warm location, I wouldn't fertilize it for a while so that the roots don't get burned.

You should see how many roots nursuries cut off of mature palms to replant them in other places!

As logn as it has some roots it will grow back (not going to promise it will grow quickly or even at all until its roots are strong again though). Some leaves might yellow, but care for it well and it will reestablish itself.

Good luck!

I agree will live i have cut off many roots and they still thrive a little slower but will regain. like walkingfrog said i wouldnt fertlize cuz the roots and get burned and damaged like an open skin wound with salt lol