Find the limit. Use l'Hospital's Rule.?

Question

lim (x^a-1)/ (x^b-1)
x->1

Answer 1

As x --> 1 numerator --> 1^a - 1 = 1 - 1 = 0
Similarly for the denominator
0 / 0 is undefined, so use l'Hopital's Rule

derivative of numerator = a x^(a-1)
derivative of denominator = b x^(b-1)

As x --> 1 a x^(a-1) --> a
As x --> 1 b x^(b-1) --> b
So limit is a / b

Answer 2

You don't need L'Hosptial's rule for this. 1 to any power is 1. So, as x -> 1, the numerator and denominator will both go to 1, and the result will be 1.