If _ x → 1 x ^ 4 - 1 x - 1 = _ x → k x ^ 3 - k ^ 3 x ^ 2 - k ^ 2 ,...

Given limits can be written as,

$\lim_{x \to 1}\dfrac{\left({x}^{2} + 1\right)\left(x + 1\right)\left(x - 1\right)}{\left(x - 1\right)} = \lim_{x \to k}\dfrac{\left(x - k\right)\left({x}^{2} + {k}^{2} + xk\right)}{\left(x - k\right)\left(x + k\right)}$
$⇒ 4 = \dfrac{3{k}^{2}}{2k}$
$⇒ k = \dfrac{8}{3}$.