Answer:
Option B
Explanation:
Concept involved $( \frac{\infty}{\infty})$ form
$\lim_{x \rightarrow \infty}\frac{a_{0}x^{n}+a_{1}x^{n-1}+.....+a_{n}}{b_{0}x^{m}+b_{1}x^{m-1}+.....+b_{m}}$
=$\begin{cases}0 ,& if n=m\\\frac{a_{0}}{b_{0}}, & if n <m \\+\infty, &if n>m and a_{0}b_{0}>0\\-\infty&if n>m and a_{0}b_{0}<0\end{cases}$
Description of Situation As to make degree of numerator equal to degree of denominator.
Sol. $\lim_{x \rightarrow \infty}\left(\frac{x^{2}+x+1}{x+1}-ax-b\right)=4$
$\Rightarrow$ $\lim_{x \rightarrow \infty}\frac{x^{2}+x+1-ax^{2}-ax-bx-b}{x+1}=4$
$\Rightarrow$ $\lim_{x \rightarrow \infty}\frac{x^{2}(1-a)+x(1-a-b)+(1-b)}{x+1}=4$
Here, we make degree of Nr = degree of Dr
$\therefore$ 1-a=0
and $\lim_{x \rightarrow \infty}\frac{x(1-a-b)+(1-b)}{x+1}=4$
$\Rightarrow$ 1-a-b=4
$\Rightarrow$ b=-4 $[ \because (-a)=0]$
