Cats having teeth taken out is fine (i have had cats with no teeth), but if they are broken you have to be careful that they dont cut their mouths and turn into abscesses, so they should always be looked at by a vet.
Good for you for taking your cat to the vet for the broken tooth. Depending on your kittys age & condition of all the teeth, your vet may recommend a full cleaning, if an extraction of the broken tooth is necessary. If it is...do it. It is well worth the expense. Please keep us updated.