QJEEAdv12Ma7991: JEE Advanced 2012 Mathematics - Class 12 JEEAdv12Ma7991 Solution: If , then

$\mathop {\lim }\limits_{x \to \infty } \left( {{{{x^2} + x + 1} \over {x + 1}} - ax - b} \right) = 4$

$\Rightarrow \mathop {\lim }\limits_{x \to \infty } {{{x^2} + x + 1 - a{x^2} - ax - bx - b} \over {x + 1}} = 4$

$\Rightarrow \mathop {\lim }\limits_{x \to \infty } {{{x^2}(1 - a) + x(1 - a - b) + (1 - b)} \over {x + 1}} = 4$

Here, we make degree of N^r = degree of D^r

$\therefore$ $1 - a = 0$

and $\mathop {\lim }\limits_{x \to \infty } {{x(1 - a - b) + (1 - b)} \over {x + 1}} = 4$

$\Rightarrow 1 - a - b = 4$

$\Rightarrow b = - 4$