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

Given: limx→∞( (x2+x+1)/(x+1) - ax - b) = 4
→ limx→∞( (x2+x+1 - ax2 - ax - bx - b)/(x+1) ) = 4
→ limx→∞( x2(1-a) + x(1-a-b) + (1-b) )/(x+1) = 4
This is possible only when
1-a = 0 → a = 1
and 1-a-b = 0 → 1 - 1 - b = 0 → b = 0
Therefore, a=1 and b=0.
However, if we consider the limit as:
limx→∞( (x2+x+1)/(x+1) - ax - b) = limx→∞( x - ax - b + (1-b)/(x+1) ) = 4
For the limit to exist, the coefficient of x must be zero, so 1-a=0 which implies a=1
Then the limit becomes limx→∞( -b + (1-b)/(x+1) ) = -b = 4, thus b=-4
Let's check:
limx→∞( (x2+x+1)/(x+1) - x +4) = limx→∞( (x2+x+1 -x2 -x +4x +4)/(x+1) ) = limx→∞( 4x+5)/(x+1) = 4
Therefore a=1 and b= -4 which is not given in options. There seems to be a mistake in the question or options.