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AIEEE 2012

JEE Main 2012 Mathematics question on Limits Continuity and Differentiability

If lim⁡x→∞(x2+x+1x+1−ax−b)=4\lim_{x \to \infty} \left( \dfrac{x^2 + x + 1}{x + 1} - ax - b \right) = 4, then
  1. A.
    a=1,b=4a = 1, b = 4
  2. B.
    a=1,b=−4a = 1, b = -4
  3. C.
    a=2,b=−3a = 2, b = -3
  4. D.
    a=2,b=3a = 2, b = 3
Show answer and solution

Correct option: B

Solution

Perform polynomial long division: (x² + x + 1)/(x + 1) = x + 0 + 1/(x+1), more carefully: x² + x + 1 = (x+1)·x + 1, so (x²+x+1)/(x+1) = x + 1/(x+1).

So the expression becomes x + 1/(x+1) - ax - b = (1-a)x - b + 1/(x+1).

For the limit as x→∞ to be finite (equal to 4), the coefficient of x must vanish: 1 - a = 0, giving a = 1.

Then the limit equals -b + 0 = 4, so b = -4. Thus a = 1, b = -4.

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