How do I determine the value of b so that each limit exists?
First you should factor the numerator:
Now, we should recognize that one of two things can happen from the term in the denominator. There will either be a removable discontinuity or an infinite discontinuity. If the denominator term will cancel with a term in the numerator, that means it will be a removable discontinuity. Otherwise it will be infinite.
Notice that at a removable discontinuity, the limit exists, but it doesn't exist at an infinite discontinuity.
So there are two values of that will cause there to be a removable discontinuity.