Continuity and Differentiability
Limits to Infinity
IIT-JEE 2012 Paper 1
Grade 12
Question:
If $\lim_{x \to \infty} \left(\frac{x^2 + x + 1}{x + 1} - ax - b\right) = 4$, then $(a, b)$ equals:
(1) $(1, 4)$
(2) $(1, -4)$
(3) $(1, 3)$
(4) $(-1, 4)$
Step-by-Step Solution
Key Concept: $\frac{x^2+x+1}{x+1} = x - \frac{x-1}{x+1} \cdot 1$. Long-divide: $x^2 + x + 1 = (x + 1)(x) + 1$, so $\frac{x^2+x+1}{x+1} = x + \frac{1}{x+1}$. Then $x + \frac{1}{x+1} - ax - b = (1 - a)x - b + \frac{1}{x+1}$. For finite limit: $a = 1$. Remaining limit: $-b + 0 = 4 \Rightarrow b = -4$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)