If I fell and broke my arm on a sidewalk that was raised by a tree root, can I sue the city?

Such city lawsuits are pretty common, especially in bigger metropolitan areas wherein miles and miles of sidewalk are under city responsibility and there are fewer and fewer workers able to fix all the reported problems. If your broken arm is a direct result of the negligence of the city who failed to reasonably eradicate this hazard, then yes, you could probably successfully sue the city. If you have documented proof that you contacted the city about this very tree root and sidewalk and that through no fault of your own (you using reasonable care) you fell, then by all means consider consulting with and hiring an attorney who can help you file the proper suit or possibility of suit.