Can I cap a broken tooth and kill the nerve to save it?

Yes, you can have a root canal performed and a crown placed to avoid an extraction IF there is enough healthy tooth structure without major infection to support this dental treatment. In some cases of severe trauma, the tooth can not be saved.

Thank you for your great question. Short answer is It depends. As long as the tooth broke above the bone level and the crack does not extend down into the root, tooth may be saved. In case there is a vertical fracture going deep into the bone, prognosis is usually poor and it is better to extract the tooth and replace it with an implant.All the best to you.

Yes- this is possible, but it depends on the situation. 
If the tooth only has a smaller chipped / broken area that hasn't come close to the nerve, it may be possible to save the tooth with a filling, onlay or crown. 
If the broken part of the tooth is deep and has exposed the nerve, and yet there is still enough tooth structure left as an adequate base for a crown, then a root canal may be done (removing the nerve, cleaning the canal where the nerve was and sealing it) and a crown may be placed.
If the broken part of the tooth has decayed or has consumed such a significant part of the tooth that a crown no longer is a viable option, then extraction of the tooth may be a better option. 
Hope this helps!