What is the limit as x -> ∞ of (x^2 + 2) / (x - 1)?

This function diverges; the limit is 'equal' to $\infty$.
Divide both the numerator and the denominator by $x$, that makes it easier.
${\lim}_{n \to \infty} \frac{{x}^{2} / x + \frac{2}{x}}{\frac{x}{x} - \frac{1}{x}} = {\lim}_{n \to \infty} \frac{x + \frac{2}{x}}{1 - \frac{1}{x}}$
We know that ${\lim}_{n \to \infty} \frac{1}{x} = 0$, so
${\lim}_{n \to \infty} \frac{x + \frac{2}{x}}{1 - \frac{1}{x}} = {\lim}_{n \to \infty} \frac{x + 0}{1 - 0} = {\lim}_{n \to \infty} \frac{x}{1} = {\lim}_{n \to \infty} x = \infty$