If _ x 0 45^x - 9^x - 5^x + 1 (k^x - 1)(3^x - 1) = 2 , then the value of k is ...

Step-by-step solution

The given limit is _ x 0 45^x - 9^x - 5^x + 1 (k^x - 1)(3^x - 1) = 2 Factorizing the numerator: 45^x - 9^x - 5^x + 1 = 9^x(5^x - 1) - 1(5^x - 1) = (9^x - 1)(5^x - 1) Substituting this back into the limit: _ x 0 (9^x - 1)(5^x - 1) (k^x - 1)(3^x - 1) = 2 Dividing the numerator and the denominator by x^2 : _ x 0 9^x - 1 x 5^x - 1 x k^x - 1 x 3^x - 1 x = 2 Using the standard limit _ x 0 a^x - 1 x = a , we get: 9 5 k 3 = 2 Since 9 = (3^2) = 2 3 , the equation becomes: 2 3 5 k 3 = 2 2 5 k = 2 k = 5 k = 5 Answer: 5