Evaluate the limit. 1. lt x-->1 (x^a-1)/(x^b-1).

1) We'll substitute x by the value 1 and we'll calcualte
the limit.

lim (x^a - 1)/(x^b - 1) = (1^a - 1)/(1^b - 1) =
(1-1)/(1-1) = 0/0

Since the result represents
an indetermination, we'll apply L'Hospital rule.

lim (x^a -
1)/(x^b - 1) = lim (x^a - 1)'/(x^b - 1)'

(x^a - 1)' =
a*x^(a-1)

(x^b - 1)' =
b*x^(b-1)

lim (x^a - 1)'/(x^b - 1)' =
lim  a*x^(a-1)/b*x^(b-1)

We'll apply the quotient rule of
the power function:

x^m/x^n =
x^(m-n)

We'll put m = a-1 and n =
b-1

x^(a-1)/x^(b-1) =
x^(a-1-b+1)

x^(a-1)/x^(b-1) =
x^(a-b)

lim  a*x^(a-1)/b*x^(b-1) = (a/b)*lim
x^(a-b)

We'll substitute x by
1:

(a/b)*lim x^(a-b) = (a/b)*1^(a-b) =
a/b

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