The values of α and β such thatlimx→∞ x2+1x−1−αx−2β=32 are

$lim_{x \to \infty} [\frac{x^{2} + 1}{x - 1} - α x - 2 β]$

$= lim_{x \to \infty} \frac{x^{2} (1 - α) - (2 β - α) x + 1 + 2 β}{x - 1}$

Since the last exists so $1 - α = 0 \Rightarrow α = 1$ . In this
case the last limit is equal to

$lim_{x \to \infty} \frac{- (2 β - α) + \frac{1 + 2 β}{x}}{1 - \frac{1}{x}} = - (2 β - α) = \frac{3}{2}$

$\Rightarrow - 2 β = \frac{3}{2} - 1 = \frac{1}{2} \Rightarrow β = - \frac{1}{4}$ .

The values of $α and β$ such that
$lim_{x \to \infty} [\frac{x^{2} + 1}{x - 1} - α x - 2 β] = \frac{3}{2} are$