Let f(x) be a polynomial function such that f(x) + f ^ ' (x) + f ^...

Let $f\left(x\right) = \left(x - 1\right)\left({x}^{4} + a{x}^{3} + b{x}^{2} + cx + d\right)$

$f\left(x\right) + {f}^{'}\left(x\right) + {f}^{''}\left(x\right) = {x}^{5} + \left(a + 4\right){x}^{4} + \left(b + 3a + 16\right){x}^{3} +$

\left(9a + 2b + c - 12\right){x}^{2} + \left(d + c + 4b - 6a\right)x + \left(c - 2b\right)$</p><p>$∵ f\left(x\right) + {f}^{'}\left(x\right) + {f}^{''}\left(x\right) = {x}^{5} + 64$</p><p>$⇒ a + 4 = 0, b + 3a + 16 = 0, 9a + 2b + c - 12 = 0, d + c + 4b - 6a = 0$</p><p>$⇒ a = - 4, b = - 4, c = 56, d = - 64$</p><p>So$\lim_{x \to 1}\dfrac{f\left(x\right)}{x - 1} = \lim_{x \to 1}\left({x}^{4} - 4{x}^{3} - 4{x}^{2} + 56x - 64\right)$</p><p>$= 1 - 4 - 4 + 56 - 64 = - 15$