If _ n → ∞ (an - 1 + n^ 2 1 + n ) = b , where a is finite number,...

$\lim_{n \to \infty}\dfrac{an ( 1 + n ) - \left(1 + n^{2}\right)}{1 + n}$

$= \lim_{n \to \infty}\dfrac{( a - 1 ) n^{2} + an - 1}{n + 1} = \infty$,

If $a - 1 \ne 0$ limit does not exist and if $a - 1 = 0$, then

$\lim_{n \to \infty}\dfrac{an - 1}{n + 1} = a = b$

$⇒ a = b = 1$