Now I've said it I want to follow with its not your fault.
Kids have so much emotion that builds, and they havent learned what to do with it yet. You can be a near perfect parent and when they get frustrated or angry they try to get that emotion out (who wouldn't, its not a nice feeling) and they are limited on how to do that. So they hit, they yell, and they bite. Its our job as parents to teach them other ways to express feelings and to get rid of those feelings.
Making them feel worse or guilty because they have these feelings will only increase the problem. They need to know its normal to have feelings, that other people (even mommy) has felt them, and thisos how we deal with it.
Then decide how you think they should express those feelings. We use words. I want to hear how she is feeling and talk through it. This won't work with younger children though, I've heard telling into or punching at a pillow helps younger children. I used to hug it out, the madder she was the bigger the hug, and we "blow out" bad feelings with a big breath of nice fresh air and we blow them away. Imagining the bad feelings floating away and turning into good feelings as they hit the clouds, or trees, etc causes an actual image for them to focus on. Distract them from hitting and biting by giving them an outlet and a different focus and they will usually calm down faster.
As parents its our job to teach them how to deal with real life things, so they are prepared for adulthood. If you don't teach them that some people can tell you what to do then they won't hold a job or listen to authorities (think teachers, police, employers). If you don't teach them what to do with anger they will forever strike out. When we don't teach our children how to make decisions and respect their choices they will forever follow the crowd and peer pressure can be dangerous. We aren't raising babies to be babies, we are raising children to be adults.