If lim x ( x 2 + x + 1 x + 1 - a x - b ) = 4 , then [2012]

Question

If $\lim_{x \to \infty} (\frac{x^{2} + x + 1}{x + 1} - a x - b) = 4,$ then [2012]

Smart Achievers · Accepted Answer

(2)

$Given: \lim_{x \to \infty} (\frac{x^{2} + x + 1}{x + 1} - a x - b) = 4$

$\Rightarrow \lim_{x \to \infty} \frac{x^{2} + x + 1 - a x^{2} - a x - b x - b}{x + 1} = 4$

$\Rightarrow \lim_{x \to \infty} \frac{(1 - a) x^{2} + (1 - a - b) x + (1 - b)}{x + 1} = 4$

$For this limit to be finite, 1 - a = 0 \Rightarrow a = 1$

$Then given limit reduces to:$

$\lim_{x \to \infty} \frac{- b x + (1 - b)}{x + 1} = 4$ $\Rightarrow \lim_{x \to \infty} \frac{- b + \frac{(1 - b)}{x}}{1 + \frac{1}{x}} = 4$

$\Rightarrow - b = 4$ or $b = - 4$ $∴$ $a = 1, b = - 4$

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