If `lim_(x to 1) (x^(3) - 1)/(x - 1) = lim_(x to k) (x^(4) - k^(4))/(x^(3) - k^(3))`, find the value of k.

Explanation

Firstly, we have `underset(x to 1)(lim) (x^(3) - 1)/(x - 1) = underset(x to 1)(lim) ((x)^(3) - (1)^(3))/(x - 1) = 3 (1)^(2) = 3`
Similarly, `underset(x to k)(lim) (x^(4) - k^(4))/(x^(3) - k^(3)) = underset(x to k)(lim) ((x^(4) - k^(4))/(x - k))/((x^(3) - k^(3))/(x - k)) = underset(x to k)(lim) (x^(4) - k^(4))/(x - k) // underset(x to k)(lim) (x^(3) - k^(3))/(x - k) = 4(k)^(3) // 3(k)^(2) = (4)/(3) k`
Now, `underset(x to 1)(lim) (x^(3) -1)/(x - 1) = underset(x to k)(lim) (x^(4) - k^(4))/(x^(3) - k^(3)) implies 3 = (4)/(3) K implies k = (9)/(4)`