I'm not clear how you can use L'Hopital and NOT get a. Differentiating both numerator and denominator, gives and using L'Hopital's rule again, which has a limit of . In order for that to be 1, b must be .
But, in order that we be able to use L'Hopital a second time, on the fraction , since the denominator goes to 0, the numerator must also- we must have . That is a= -b.