TITLE: How important are the role of asymptotes in a hyperbola?
QUESTION [7 upvotes]: Let A be the hyperbola with the equation $\displaystyle \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, where $a$ is the $x$-intercept and $b$ is the $y$-intercept.
Given this it can be calculated that the lines $\displaystyle y=\frac{b}{a}\cdot x$ and $\displaystyle y=-\frac{b}{a}\cdot x$ are both asymptotes to the hyperbola.
The question is simply: are these asymptotes significant? And if so, why?

REPLY [7 votes]: Yes, the asymptotes tell you the global behavior of the hyperbola, and in general it is extremely useful to be able to explain the global behavior of something complicated using simpler things that you already understand well (such as linear functions).
Specifically, they tell you that a hyperbola becomes linear far away from its center.