UPSC NDA I 2015 Mathematics Paper

Concept:For rational functions, the limit at infinity is determined by the ratio of leading coefficients when the degrees of numerator and denominator are equal.Explanation:We are given $\underset{x \to \infty}{\lim} \left( \frac{2+x^2}{1+x} - Ax - B \right) = 3$.From earlier steps, $A = 1$. Substitute $A = 1$ into the limit:$\underset{x \to \infty}{\lim} \left( \frac{2+x^2}{1+x} - x - B \right) = 3$.Combine into a single fraction:$\frac{2+x^2 - x(1+x) - B(1+x)}{1+x} = \frac{2+x^2 - x - x^2 - B - Bx}{1+x} = \frac{-(1+B)x + (2-B)}{1+x}$.So the limit becomes $\underset{x \to \infty}{\lim} \frac{-(1+B)x + (2-B)}{1+x} = 3$.Divide numerator and denominator by $x$:$\underset{x \to \infty}{\lim} \frac{-(1+B) + \frac{2-B}{x}}{\frac{1}{x} + 1} = \frac{-(1+B)}{1}$.Thus $-(1+B) = 3$, which gives $1+B = -3$, so $B = -4$.Answer:$B = -4$