If _ x 1 x^4-1 x-1 = _ x k x^3-k^3 x^2-k^2 , then the value of K is

Step-by-step solution

Evaluate the first limit: _ x 1 x^4-1 x-1 = _ x 1 (x-1)(x^3+x^2+x+1) x-1 = _ x 1 (x^3+x^2+x+1) = 1+1+1+1 = 4 . Evaluate the second limit: _ x k x^3-k^3 x^2-k^2 = _ x k (x-k)(x^2+xk+k^2) (x-k)(x+k) = _ x k x^2+xk+k^2 x+k = k^2+k^2+k^2 k+k = 3k^2 2k = 3k 2 . Equate the two limits: 4 = 3k 2 . Solve for k : 3k = 8 k = 8 3 . Answer: 8 3