Asymptotes of a hyperbola

Consider the hyperbola in standard form:

$\displaystyle \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $.

My textbook tells me that the lines with the equations

$\displaystyle \frac{x^2}{a^2} - \frac{y^2}{b^2} = 0 $, or $\displaystyle y = \pm \frac{b}{a} x$

are the asymptotes of the hyperbola.

In general terms, I understand that asymptotes are those lines to which the curve tends as $\displaystyle x $ or $\displaystyle y \to \pm \infty $, but I can't see how that has been applied here to the hyperbola.