With Grizzly or Polar bear, you want your lead to have high antimony fraction, so the bullet is hard and penetrates. Most Black bears are not so aggressive. Sun bears (Asia) are even more aggressive but not so big and tough, rather like an oversized wolverine!
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Yesterday someone ran over a black bear near here. He only weighed 225 and the game people said he was 4 years old.
In this area, black bears are not real common but they are here. My neighbor has seen one several times during his daily walks.
I know a .22lr would not be adequate for bear, especially from a 4 inch barrel. On the other hand I once watched my daddy kill a 550 lb. sow with a single shot from a .22 rifle. The cartridge was a .22 short. The sow dropped instantly.