If $\lim_{x \to \infty} \left( \frac{x^{2} + x + 1}{x + 1} - ax - b \right) = 4$, then

If lim_(x → ∞) ( (x² + x + 1)/(x + 1) - ax - b ) = 4, then

$\text{Given: } \lim_{x \to \infty}\left(\dfrac{x^2 + x + 1}{x + 1} - ax - b\right) = 4$

$\text{To find: } a,\ b \text{ and the correct option}$

$\dfrac{x^2 + x + 1}{x + 1} = x + \dfrac{1}{x + 1}$

$\lim_{x \to \infty}\left(x + \dfrac{1}{x+1} - ax - b\right)$

$= \lim_{x \to \infty}\left((1-a)x + \dfrac{1}{x+1} - b\right)$

$\text{Finite limit} \Rightarrow 1 - a = 0 \Rightarrow a = 1$

$\lim_{x \to \infty}\left(\dfrac{1}{x+1} - b\right) = -b = 4$

$\Rightarrow b = -4$

$a = 1,\ b = -4 \Rightarrow \text{option (2)}$

$\boxed{a = 1,\ b = -4}$