To find the value of the limit
, we can start by noting that both the numerator and the denominator approach zero as
x approaches 1 . This situation allows us to use L'Hôpital's rule, which is applicable when we have an indeterminate form of type
.
By L'Hôpital's rule, we can take the derivatives of the numerator and the denominator and then evaluate the limit:
We can simplify the fraction inside the limit:
Now evaluate the limit as
x approaches 1 :
⋅15==Thus, the value of the limit is:
Therefore, the correct answer is Option C:
.
Alternate Method
We can use the factorization
an−bn=(a−b)(an−1+an−2b+...+abn−2+bn−1)
to simplify the given expression.
Now, we can cancel out the common factor of
(x−1) :
Since the denominator is not zero when
x approaches 1 , we can simply substitute
x=1 to find the limit:
Therefore, the correct answer is Option C.